It is important to match the behaviour of the stochastic model against the concentration curves predicted from ODE models. Here I show that there is a qualitative match between the rates and concentration profiles predicted by ODE models of template replication and the behaviour of the stochastic model. However once more realistic elongation side-reactions are permitted in the full stochastic model, e.g. p-bond formation on s-bonds and templated p-bond formation allowing elongation, that qualitative differences arise between my stochastic model and simple ODE models that do not account for these more realistic side-reactions. In the second part I show that the rates and extents of dissociation of double stranded oligomers is in qualitative agreement with the behaviour of real olgiomers, e.g. longer oligomers dissociate more slowly at the same temperature than shorter oligomers, and to a lesser extent. The mathematica file used to plot validation results can be found here.

P-bond Dynamics Validation

A simple model of the minimal template replicator is shown below (using Cellerator format).

The binding of substrate (A and B) to the template C has K = 1, and the dissociation of C2 double strands has K = 0.1, i.e. the double strands are 10 times less likely to dissociate than the single strands, being twice as long in length. The rate of phosphodiester bond formation ( ABC -> C2) is orders of magnitude slower than the hydrogen bond mediated association and dissociation events. The behaviour of the system is shown below.

The behaviour predicted by this simple model is compared to the results obtained using my stochastic template simulator. The stochastic simulator was initialized with 5 double stranded GCGC 4-mers, and 95 double stranded GC dimers, in a very small volume, giving an initial total concentration of 4-mers = 0.001M and 2-mers = 0.01M. The system was simulated at a temperature of 275.0 K. The behaviour of the experiment is shown below. Only a few p-bond breakage events occured. The results do not compare well to the simple minimal replicator model since side-reactions result in the production of 6 and 8-mers, and this is not accounted for by the ODE model. Also p-bond breakage results in production of dimers broken off from oligomers. Furthermore, spontaneous p-bond formation on s-bonds is permitted. These side-reactions complicate matters. None of these features are included in the simple ODE model, but it is reasonable to suppose that they arise in real nucleic acid chemistries.

Concentration is given in M, and time in seconds. The total length of the experiement is of approx. 11 days. The joined lines show an ODE model that has the same ligation rate as the stochastic simulation [see below]. The 4-mers grow to a maximum concentration, but are quickly used to make even longer molecules.

The gaps in the graph represent p-bond breakage events, that occur rarely. Due to the small number of samples of microstates that can be made, if a configuration of templates that could form a p-bond is not found, then a p-bond breakage event is executed. However, the fact that further p-bond formation events could occur in the time between the present and the time at which the p-bond breakage event is expected to occur, are not considered. Thus, p-bond breakage events are over-estimated. However, I did not worry about this since it would only hinder replication, not aid it.

The equations for the ODE model used to plot the lines above are shown below.

The next experiment attempts to force the stochastic model to match the ODE model by preventing realistic side-reactions that actually do occur. These side-reactions were blocked by Z & O in their experiments with GCGC replicators. This should result in a much closer match to the idealized ODE model. The Z and O side-inactivation is simulated by simply preventing the formation of a p-bonds that would produce a template that is longer than 3 p-bonds in length, i.e. in the calculation of p-bond formation propensity, I ensure that the propensity of p-bond formation is zero if there is more than one p-bond on the left or the right of the position at which the p-bond could form. S-bonds are allowed to form at the same rate, but there is zero rate at which p-bonds form upon s-bonds. The following results are observed.

The results show that no strands longer than 4-mers are produced as expected by our prevention of side-reactions. There is a 'rapid' formation of GCGC mers that follows a curve much more like that predicted by the ODE. After the short period in which p-bonds are created there is a long phase of exponential decay where p-bonds are broken. The graph above shows the long phase and the rapid growth phase as an inset. You can see that the curve of the rapid growth phase is much closer to the curve predicted by the ODE than in the more realistic case where elongating side-reactions are permitted. The graph below shows the predicted curve for GC and GCGC. However, it is not fitted directly to the stochastic model since there are several p-bond breakage events at the start of the stochastic simulation that results in the growth phase being shifted to the right. This is perhaps because insufficient time was given for h-bond dynamics to reach equilibrium before the first p-bond event was calculated in the stochastic model. Typically 0.0005 seconds are allowed for equilibrium of ds/ss strands is reached. As we see below in the h-bond dynamics validation section, for longer strands this may be insufficient time for ds/ss equilibrium to be reached.

 

Next I compare CGCG to GCGC in the idealized model. A reactor was initialized with the same concentrations as above, but with CGCG instead of GCGC. There is also growth of CGCG in the idealized case where p-bonds are only allowed to form where they would produce CGCG, and no other product.

The same run as above was conducted in non-idealized conditions, i.e. where p-bonds can form between any nucleotide, and where p-bonds can form on s-bonds.

The behaviour was quite similar to GCGC, except that there was an earlier peak in CGCG content. This shows that CGCG was more rapidly elongated into CGCGCG than was GCGC.

H-Bond Dynamics Validation

Above I examined the p-bond formation dynamics, and showed that they were an qualitative agreement with the dynamics expected from ODE models. Finally, I observe the kinetics of h-bond dynamics. In particular I observe the rate of dissociation of strands from the double stranded state at different temperatures. Longer strands should show a longer time to reach their equilibrium ds/ss ratios than shorter strands.

Below I show the rate at which single strands are produced when double stranded 4-mers of GCGC are placed in a reactor at different temperatures with initial concentration 0.01mM.

You can see that at higher temperatures the rate of single strand formation is more rapid, as is the total extent of single strands produced. The graph below is the same as that above, except I superimpose the curve for 5-mers of GCGCG at 295 K.

You can see that the time for dissociation of 5-mers is longer, and the extent of dissociation is less, than the 4-mers at the same temperature (295K). Below I show the same experiment for 6-mers of GCGCGC.

A similar pattern is observed. Below I show the dissociation curve for 8-mers at 330K and 10-mers at 350K.

The simulation is able to capture the melting temperatures and the temporal differences in dissociation rates exhibited by templates of different lengths.

 

Validation 2: A Similar Experiment to Dirk Sievers and Guenter Von Kiedrowski (1994).

The following graph shows a system initialized with 5 ds 5'- CCGCGG - 3' templates and 95 5'- *CCG- 3' and 95 5' - GGC* - 3' ss building blocks. The building blocks have been treated so that the ends marked with a star cannot form p-bonds. The system has a QV = 10000, therefore this is equivalent to a concentration initially of 1mM template and 10mM building blocks each. A lower concentration can also be employed. The temperature is set to 300K. In an initial experiment, p-bonds were allowed to form every 0.00005 seconds of h-bond dynamics. This resulted in the following pattern of growth of CCGCGG templates. The simulation code used to generate these results is downloadable here.

When I tried this with the same volume used by Sievers and Von Kiedrowski, i.e. 10 x the previous volume, so that the template is present at concentration 0.1mM and the building blocks at 1mM, the strands were completely dissociated at this temperature. This suggests that our setting of polymer collision rates has not been appropriately scaled. It is useful to measure the melting temperatures of CCGCGG and GGC obtained with the current settings, since it may well be the case that there is merely a linear displacement of the Tms as a function of concentration, due to an improperly set polymer association frequency. For example, volume can be increased without any effect if the BIMOLECULAR constant is increased also from 5000 to 5000x100. The results at this lower 'concentration' exhibiting the same behaviour. Below I show some experiments with these settings and the longer standard h-bond dynamics time of 0.0005 seconds between p-bond events. The next thing to do is try this with different initial concetrations of template. Note, that no s-bond formation at all is permitted, so these are the dynamics of pure templated p-bond formation.

Compare both of these above results where CCGCGG is provided with an experiment where no CCGCGG is provided and s-bond formation is again fobidden.

In the following experiment, although no CCGCGG is provided, s-bond formation is permitted. In this case however, I have not yet simulated end - blocked building blocks.

 

 

 

 

Tidal Cycling Experiments

Further experimental results from tidal cycling experiments and controls can be found here.