University of Groningen Enabling Darwinian evolution in chemical replicators Mattia, Elio

University of Groningen

Enabling Darwinian evolution in chemical replicators

Mattia, Elio

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2017

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Mattia, E. (2017). Enabling Darwinian evolution in chemical replicators. Rijksuniversiteit Groningen.

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2.

EXPONENTIAL

REPLICATION

Self-replicating molecules are likely to have played a central role in the origin of life. Most scenarios of Darwinian evolution at the molecular level require self-replicators capable of exponential growth. Yet only very few exponential self-replicators have been reported to date and general design criteria for exponential replication are lacking. Here we show that a peptide-functionalised macrocyclic self-replicator exhibits exponential growth when subjected to mild agitation. The replicator self-assembles into elongated fibres of which the ends promote replication and fibre growth. Agitation results in breakage of the growing fibres, generating more fibre ends. Our data confirm the role of a mechanism in which mechanical energy promotes the liberation of the replicator from the inactive self-assembled state, thereby overcoming self-inhibition that prevents the majority of self-replicating molecules developed to date from attaining exponential growth.

The work for this chapter was realized in collaboration with Mathieu Colomb-Delsuc, who carried out the experimental studies and part of the data analysis.

Parts of this chapter have been published:

Mathieu Colomb-Delsuc, Elio Mattia, Jan W. Sadownik & Sijbren Otto. Exponential self-replication enabled through a fibre elongation/breakage mechanism. Nat. Commun. 2015, 6, 7427.

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n In Chapter 1, we introduced replicators and illustrated the inhibiting role of duplex formation during replication. In this chapter, we report experimental and computational results on exponential replication. Our studies indicate that a growth/breakage mechanism, which does not involve duplex formation, allows for replicating molecules to grow exponentially.

The importance of exponential replicators resides in its peculiar evolutionary traits. As shown in Chapter 1 (Section 1.2.1 and Figures 1.2.1 and 2.1.1a), most replicators suffer from duplex formation. Such replicators generally display parabolic growth, whereas exponential and hyperbolic replicators display phenomena such as survival of the fittest or survival of the common (Chapter 1, Section 1.2.2). In far-from-equilibrium conditions, exponential replicators are particularly prone to survive destruction reactions and attain dynamic kinetic stability and, ultimately, evolve (Chapter 1, Sections 1.2.3 and 1.2.4). There are very few experimental systems that are deemed to display exponential replication and to our knowledge a general principle by which exponential replication can be obtained has not yet been documented.

2.1. EXPERIMENTAL FRAMEWORK

The experimental system that the work in this thesis builds upon is based on our previous observations on the emergence of replicators from dynamic combinatorial libraries1 _{arising from thiol-functionalized peptide building blocks} such as 12,3 _{(Figure 2.1.1b). Here we demonstrate experimentally that the} replication process in our system is consistent with exponential growth. We then offer experimental insights into the mechanism of replication: a key role for fibre breakage is evident from the fact that the stirring rate influences both the rate of replication and the concentration of fibre ends. We demonstrate a direct correlation between average fibre length and replication rate. Furthermore, we show that fibre length distributions remain constant during replication, which is necessary in order to achieve exponential replication. Finally, we extensively confirm computationally that our hypothesized fibre elongation/breakage mechanism results in exponential replication and that the fibre breakage step is required for exponential growth. Air oxidation of an aqueous solution of 1 gives rise to a continuously exchanging pool of disulfide macrocycles (Figure 2.1.1b). Initially, interconverting cyclic disulfide trimers and tetramers dominate but, upon agitation, larger macrocycles such as hexamers nucleate and subsequently self-assemble into fibres (Figure 2.1.1c) and become the major product in solution (Figure 2.1.1d), eventually resulting in the total depletion of the smaller macrocycles. Hexamers are only stable in the aggregated form. Note that monomer, trimer or tetramers do not assemble into fibres. Hexamer growth is sigmoidal and seeding experiments confirmed that the growth process is autocatalytic, i.e., hexamer fibres are

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replicating. Shear stress is crucial for exponential replication: in a non-agitated library trimers and tetramers are the only significant products, and no fibres could be observed by TEM analysis of libraries dominated by those two species. The hexamer replicator only emerges when libraries are mechanically agitated. A mechanism for the replication process was proposed that involves two steps: fibre elongation by growth of the fibres from their extremities and fibre breakage due to mechanically-induced shear stress, which converts primary nuclei into secondary ones. Fibre growth by elongation may take place by sequestration of hexamer macrocycles from the macrocycle equilibrium, or through a mechanism in which fibre ends catalyse the formation of the hexamer macrocycle.

Experiments which yield different macrocycle distributions under different initial conditions have confirmed the kinetic nature of the self-replication process.2 Multiple building blocks with diverse peptide sequences are capable of forming replicating fibres of variable macrocycle size.3

Furthermore, it has been shown that a system containing multiple building blocks in the same mixture can display sequential emergence of different families of replicators.4

Figure 2.1.1 | Experimental framework. (a) Traditional mechanism based on template-directed ligation of two replicator precursors. (b) Oxidation of building block 1 containing two thiol

functionalities leads to a mixture of interconverting macrocycles. (c) Self-assembly of hexamers 16

results in the formation of fibres. Fibre fragmentation results in doubling of the number of fibre ends. (d) Species distribution as a function of time of a non-seeded DCL made from 3.8 mM 1 in 50 mM borate buffer pH 8.1 stirred at 1500 rpm.

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n Parallels exist between this mechanism and the nucleation-growth mechanisms of amyloid fibres, in which mechanically induced breakage can also play a role5-9_{. The} hierarchical self-assembly process (monomers organizing into hexamers that then stack into fibres) is somewhat reminiscent of the fully non-covalent assembly of rosettes made from melamine derivatives and cyanuric acid which have been proposed as prebiotic precursors to RNA analogues10,11_{. However, neither amyloids} nor melamine rosettes involve covalent bond formation or self-replication.

2.2. EXPERIMENTAL RESULTS

We performed a series of experimental studies on the system presented above and in Chapter 1. We analyzed the order of the reaction in the replicator and established its exponential nature. We also carried out mechanistic studies in order to determine the role fibre breakage plays in the replication process.

2.2.1 Determination of the replication order in the replicator

In Chapter 1, Section 1.2.2, we introduced replication kinetics regimes. In the absence of replicator destruction, the rate law for replication, i.e., Equation 1.3 of Chapter 1, simplifies to:

𝑑𝑑[𝑅𝑅]

𝑑𝑑𝑑𝑑 = 𝑘𝑘𝑅𝑅[𝐹𝐹]𝑓𝑓[𝑅𝑅]𝑟𝑟 (2.1) In order to determine the order in replicator r, Equation 2.1 may be re-written as:

log𝑑𝑑[𝑅𝑅]_{𝑑𝑑𝑑𝑑} = log 𝑘𝑘𝑅𝑅+ 𝑓𝑓 log[𝐹𝐹] + 𝑟𝑟 log[𝑅𝑅] (2.2)

During the initial phase of growth, the concentration of food molecules F is

approximately constant, hence log kR + f log[F] is constant. The replication order r can be determined from the slope of a plot of the initial rate of replication (log d[R]/dt) versus log[R]. We determined the initial rate of replication for different

replicator concentrations through a set of seeding experiments. A series of identical samples rich in monomers, trimers and tetramers of 1 was prepared and seeded with different amounts of a stirred solution of pre-formed hexamer fibres. The initial rate of increase of the hexamer concentration was determined by UPLC analysis (under the employed chromatographic conditions, fibres dissociate into their constituent hexamer macrocycles, which can be quantified from their UV-Vis absorbance).

The resulting plot of the initial rate of replication versus hexamer seed concentration is shown in Figure 2.2.1, which allowed us to determine a value for the order in replicator r of 0.996 ± 0.166. The error bars in Figure 2.2.1 represent one

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point) and the uncertainty in r is a standard deviation based on the complete set of

seeding experiments. Thus, within the experimental error, this system appears to be capable of exponential replication. However, given the experimental error on the value of r we cannot exclude that replication is sub-exponential based on these

experiments alone. In order to obtain additional evidence for exponential replication we performed a detailed investigation to establish the replication mechanism. We then calculated the exact value of r associated with this replication

mechanism through computer simulations (which are not subject to experimental errors).

Figure 2.2.1 | Experimental determination of the order in replicator. Initial replication rates against initial replicator concentration. Data points correspond to seed concentrations (expressed as

concentration of 16) of 31, 63, 95 and 127 µM, respectively. The error bars on the data points

correspond to one standard deviation for each seed concentration and the error on the slope is the standard deviation based on the complete set of measurements.

2.2.2 Mechanistic insights into the elongation/breakage process

In order to ascertain the validity of our postulated mechanism we further investigated our system experimentally. We confirmed two points:

1. fibre breakage plays a fundamental role in replication, as shown by using transmission electron microscopy (TEM) and verifying that shorter fibres produced by faster stirring result in more efficient replication at equal total concentrations,

2. there is the linear relationship between fibre end concentration and replicator concentration, as shown by monitoring the evolution of average fibre length along the replication pathway and confirming that it does not vary substantially.

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n First, the effects of stirring on the average fibre length and on the rate of replication were investigated. Fibre length distributions were determined using transmission electron microscopy (TEM) for identical samples of pre-formed hexamer fibres that were subjected to different stirring rates. As expected, the average fibre length decreases substantially with increasing stirring rate, ranging from 745 nm at 200 rpm to 97 nm at 1500 rpm (Figure 2.2.2a). Additionally, higher stirring rates also result in more efficient replication. Figure 2.2.2b shows the change in replicator concentration with time for different libraries stirred at rates ranging from 200 rpm to 1500 rpm: at higher stirring rates, hexamers emerge faster than at lower rates of stirring.

The two previous results together imply that in systems with shorter fibres replication is more efficient; this is also evident from Figure 2.2.2c where t50, i.e., the time it takes for the hexamer to represent 50 % of the total library material, is plotted against the average fibre length (values obtained from the data in Figure 2.2.2b). Average fibre length and t50 do indeed appear to be correlated, as expected

for a mechanism where fibre growth takes place from fibre ends; i.e. when the same quantity of replicator is distributed over a larger number of shorter fibres (hence, more fibre ends), replication is more efficient than when it is distributed over a smaller number of longer fibres (hence, fewer fibre ends).

Figure 2.2.2 | Influence of the stirring rate on fibre length and replication kinetics. (a) Average fibre length for different stirring rates. (b) Kinetics of replicator growth at various stirring rates (lighter to darker blue: 200, 400, 800, 1000 and 1500 rpm). (c) Time needed for the replicator

to represent 50% of the library material (t50), as a function of the average fibre length; the line

represents a linear fit of the data.

Finally, we also studied whether the average fibre length changed during the replication process. In fact, in a mechanism where replication takes place at the fibre ends, for replication to be exponential (i.e., r = 1), the average fibre length should not vary with time during replication. The concentration of fibre ends is only directly proportional to the total replicator concentration if the average fibre length is constant during replication. Note that, at a given amount of replicator, the concentration of fibre ends is independent of the fibre length distribution. This fact

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follows directly from the definition of the average fibre length which equals the sum of the lengths of all fibres divided by the number of fibres. The first term is constant and determined by the replicator concentration (assuming all replicator to be incorporated into the fibres) while the second term equals half the number of fibre ends. We monitored product distribution by UPLC (Figure 2.2.3a) and the average fibre length by TEM (Figure 2.2.3b) over the course of the replication process in a partially oxidised library containing mainly monomer, trimer and tetramer which was seeded at t = 0 min with pre-formed hexamer fibres and stirred at 1200 rpm.

Figure 2.2.3 | Change in replicator concentration and average fibre length during replication. (a) Change in product distribution with time of a library made from building block 1 (3.8 mM) stirred at 1200 rpm composed of 1 and cyclic trimer and tetramer seeded at t = 0 min with 20% pre-formed hexamer fibres (monomer concentration as black squares, trimers as green downward triangles, tetramers as red circles, hexamers as blue upward triangles). (b) Average fibre length (blue squares) and associated standard deviation of the fibre length distribution (green circles) of this seeded library as a function of time (determined by TEM).

Figure 2.2.3b shows that the average fibre length is essentially constant, while the total fibre concentration increased approximately fourfold (21% to 75%). During replication, the system therefore reaches a dynamic stationary state in fibre length in which fibre elongation is balanced by mechanically-induced fibre breakage. Constant fibre length suggests a direct proportionality between replication rate and concentration of fibre ends, a prerequisite for exponential replication with r=1.

2.3. COMPUTATIONAL RESULTS

In order to corroborate that our proposed elongation/breakage mechanism gives rise to exponential growth of hexamer fibres, the replicating system was also studied computationally and the order in replicator was determined through

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n numerical simulations that are unencumbered by experimental uncertainty. The exchange reactions in solution, as well as the fundamental steps for replication according to our hypothesis, i.e., fibre nucleation, elongation and breakage, were included in a kinetic model (Figure 2.3.1), which was studied under a set of parameters. We used this model to address two key questions: Firstly, does the elongation/breakage mechanism result in exponential replication, and if so, can we obtain a more accurate estimate of the order in replicator r than is possible

experimentally? Secondly, is breakage necessary for exponential replication? Related to this second question we also set out to determine the value of r in a model

that lacks breakage, but is otherwise identical to the original model.

The kinetic scheme that has been employed as the framework for our numerical simulations is depicted in Figure 2.3.1 and the simulations are described in further detail in this Section 2.3 (the source code is provided in Appendix A, Section A.1). It considers non-assembled species and self-assembled fibres of varying length, along with exchange processes and fibre elongation and breakage steps.

Figure 2.3.1 | Computational model of the replicating system. Fibre elongation and fibre breakage can be toggled on or off in order to study the role of these pathways in the replication process.

2.3.1 Modelling of the elongation/breakage mechanism

The reaction network depicted in Figure 2.3.1 considers the main species detected experimentally in the library solutions, including monomers, linear dimers, cyclic trimers, tetramers and, notably, hexamers, together with fibres made from stacks of hexamers, up to an arbitrary maximum length (of 10 or 100 hexamers stacked in a single fibre). Concentrations, in arbitrary units, are monitored against time for each of these species, including fibres for each possible length up to the maximum allowed length. Processes in the system fall in two categories. First, oxidation reactions, such as linear dimer formation from monomers, cyclic trimer formation from monomers and linear dimers, cyclic tetramer formation from linear dimers and cyclic hexamer formation from linear dimers and cyclic tetramers. Second, fast equilibration processes among the oxidized species, i.e. between cyclic trimers, tetramers and hexamers. The wiring of the reaction network, in absence of experimental evidence, is partly arbitrary, i.e. it is not possible to be certain that, e.g. tetramers are formed by direct oxidation of two dimers rather than oxidation of

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a monomer and a trimer; however, equilibria in solution ensure that different underlying pathways result in the same kinetic outcome when focusing on replication behaviour. Fibres nucleate from individual hexamers; they can be elongated from trimers, transforming a fibre of length n (number of stacked

hexamers) into a fibre of length n+1. The choice of growth from trimers is also

arbitrary; however, trimers and tetramers also equilibrate fast. Finally, a fibre breakage process ensures that longer fibres are converted into smaller fibres, in the process active due to shear stress in our experimental systems. Only breakage of fibres in half (or breakage of (2n+1)-long fibres into n-fibres and n+1-fibres) was

considered. Short fibres up to a threshold length (5 hexamers for simulations with maximum length of 10 hexamers and 10 hexamers for simulations with maximum length of 100 hexamers) were not allowed to break. Beyond the threshold length the breakage rate was set to increase linearly with fibre length, from zero probability of breakage at the next time step at the threshold length up to total certainty of breakage at the next time step for fibres of maximum length.

2.3.2 Elongation/breakage mechanism: main results and discussion

As shown in Figure 2.3.2a, simulations with the complete elongation/breakage mechanism were able to qualitatively reproduce the behaviour of the experimental system, where monomers are oxidized to trimers and tetramers via the formation of dimers (not shown for simplicity), before replication takes over and consumes the smaller macrocycles through the sigmoidal growth of hexamers species (Figure 2.1.1d). The behaviour shown in Figure 2.3.2a is generally displayed in the experiments during the course of three weeks.

We determined the order in replicator r by using Equation 2.2, under the

approximation that the term log kR + f log[F] is constant, which is only valid if the concentration of food molecules F is constant. In the present model, the concentration of food molecules F was kept constant by fixing the concentrations of trimers and non-assembled hexamers to a constant value throughout the simulation (equivalent to a system in which any consumed food molecules are instantly replenished from the surroundings). Figure 2.3.2b shows the central region of a plot of the logarithm of replication rate against the logarithm of replicator concentration (see Figure 2.3.3a for the complete plot and further details). A linear least squares fit of the data yielded an order in replicator of r =

1.0000, confirming that a fibre elongation/breakage mechanism indeed results in exponential replication. Note that obtaining an observed order in replicator of 1 is only possible under conditions in which the production of replicator through the uncatalysed background reaction is negligible. For our family of self-replicators, the background reaction is extremely slow: in the absence of agitation no significant amount of replicator 16 forms, even after 1 month.

Finally, we also monitored the fibre length distributions over time during the simulations. As shown in Figure 2.3.2d, after a short initial transient phase, the

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n length distribution remained constant throughout the replication process, similar to the behaviour we observed experimentally (Figure 2.2.3b).

2.3.3 Breakage-free mechanism: main results and discussion

In order to further ascertain the role of fibre breakage in the nature of the replication process, the system was also studied computationally under breakage-free conditions, i.e., by deactivating the corresponding breakage pathway. Interestingly, the results, shown in Figure 2.3.2c (the full plot is shown in Figure 2.3.3b), display a replication order close to 0.5, i.e., the value commonly associated to self-replicators that suffer from self-inhibition through duplex formation, as discussed in Chapter 1, Section 1.2.2.

Theoretical work by Stadler on replicators that form triplexes and do not involve mechanical breakage concluded that also in these systems product inhibition leads to an order in replicator of 0.512 _{(see Section 2.3.3 for further details). This} resemblance in order in replicator between our system and the duplex or triplex systems is coincidental, however; it has a different origin than the “square root law” of autocatalysis typically associated with replicator dimerization. It may be rationalised as follows: In a mechanism in which new fibres nucleate continuously at a constant rate, but no fibres break, the rate of replication increases linearly with time as it depends on the number of fibre ends (d[R]/dt ∼ t). Furthermore, under

these conditions it can be shown that the concentration of replicator increases proportionally to t2 _{([R] ∼ t}2_{). Figure 2.3.4 illustrates the breakage-free mechanism} and its consequences. Therefore, without fibre breakage, the rate of replication is proportional to the square root of the replicator concentration (d[R]/dt ∼ [R]0.5_). Thus, the simulations reveal that in the absence of fibre breakage no exponential growth is obtained, highlighting the crucial role of fragmentation in exponential replication.

2.3.4 Replication orders: summary

The plots in Figures 2.3.2b,c are obtained from the full plots of the logarithm of replication rates against the logarithm of replicator concentration along full numerical simulations shown in Figure 2.3.3a,b, where models involving, or excluding, fibre breakage, respectively, are considered. An autocatalytic “replication” region in the central portion of the latter graphs is surrounded by two regions in which additional effects are noticeable, the first one due to a (zero-order) nucleation rate being significant at lower replicator concentrations (“nucleation” region, with a slope smaller than the one of the main central replication region) and the second one due to the artificially imposed constant food supply not any longer capable of sustaining exponential replication at higher replicator concentrations (“saturation”, flat region, where the amount of replicator formed is limited either by the constant food supply at every time step or by the maximum fibre length being reached, illustrated in Figure 2.3.5). The central regions are the ones of interest in order to determine rand are the ones reproduced in Figures 2.3.2b,c.

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Figure 2.3.2 | Computational studies on the growth/breakage mechanism. The numerical simulations show distinct kinetics in the presence of the breakage mechanism and in the absence thereof. (a) Typical kinetics observed in the numerical simulations of the growth/breakage mechanism (monomers in black, trimers in green, tetramers in red and hexamers in blue). (b) Computational determination of the order in replicator r in the case of a growth/breakage mechanism. (c) Computational determination of the replication order in the case of an elongation-only (breakage-free) mechanism. (d) Fibre length distribution as a function of time. After an initial transient regime, replication occurs at a steady-state length distribution.

Note that the order in replicator was not built into our simulations but is a true

outcome of these simulations. In experimental systems and also in our simulations there are two processes that impact on the overall observed order in replicator. The first process is the uncatalysed formation of replicator. This process is not autocatalytic and is therefore zeroth order in replicator.

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Figure 2.3.3 | Computational results: the growth/breakage mechanism vs. removal of breakage. Three regions named “nucleation”, “replication” and “saturation” are identified. During nucleation, the zero-order nucleation process is significant relative to the replication process. Replication involves instead autocatalytic growth of hexamers; this is the region used for data fitting in order to determine the replication order. Saturation is due to different reasons in the two illustrated cases. (a) In the case of a growth/breakage mechanism, saturation is due to the constant building block supply being no longer sufficient to sustain exponentially faster replication. (b) In the case of an elongation-only (breakage-free) mechanism saturation is due to reversal to linear growth due to reaching maximum fibre length (Figures 2.3.4 and 2.3.5). This does not happen with active breakage, as the latter ensures continuous shortening of longer fibres.

Figure 2.3.4 | Evolution of the fibre population without the fibre breakage mechanism. Evolution of the fibre population at different time steps in numerical simulations which do not include the breakage mechanism (e=1, n=1, constant trimer supply). The number of elongation sites responsible for replication increases linearly with time, whereas the total hexamer count increases with the square of time, thereby making replication rate depend upon the square root of the total replicator concentration (r=0.5), at sufficiently high replicator concentrations.

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Figure 2.3.5 | Evolution of the fibre population without the fibre breakage mechanism, after maximum allowed fibre length has been reached. Evolution of the fibre population at two different time steps in numerical simulations which do not include fibre breakage (e=1, n=1, constant trimer supply), after the maximum allowed fibre length has been reached. The growth rate is now independent of total hexamer concentration.

The second process is autocatalytic and it is the order in replicator in this process that is relevant for evolutionary scenarios. The overall replication order that is determined experimentally or in our simulations can be regarded as a weighted average of the two processes. In the limiting case that all “replicator” is produced through the uncatalysed pathway the order in replicator will be zero. In the other limiting case that all replicator is produced through the autocatalytic pathway, the order in replicator may be exactly 1 (or exactly 0.5 in the case of parabolic replication). In intermediate situations, an order between 0 and 1 (or 0.5) is obtained. In fact, the observed order in replicator changes during the replication process. If one starts off the replication reaction without any replicator the first molecules of replicator can only be produced through the uncatalysed pathway and the process will be zero-order in replicator at t = 0. As more replicator gets produced

the autocatalytic pathway will become increasingly efficient, while the uncatalysed pathway will not increase in efficiency. Thus, as the reaction progresses, the order in replicator will converge on the order of the actual replication reaction. This is exactly the behaviour we observe in our simulations, as shown in Figure 2.3.3: in what we have termed the nucleation phase the observed order in replicator is smaller than 1 (converging on zero for t = 0), while during the more advanced stages

of replication (the replication phase) the amount of replicator produced by the uncatalysed pathway becomes increasingly insignificant compared to the amount produced autocatalytically and therefore the order in replicator converges on 1. In

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n between the two phases we observe replication orders that change from 0 to 1. Note that our experimental system is quite unique in that the uncatalysed formation of replicator is very slow. In the absence of fibre breakage (no agitation) we do not observe significant quantities of replicator over a period of 1 month. Thus, the uncatalysed pathway contributes very little to the overall replicator production as soon as self-replication has started.

Figure 2.3.6 | Fibre length distribution in the replication and saturation phases, with breakage. During replication, the fibre length distribution remains constant, but changes in the saturation regime. (a) Fibre length plot including the three different regions of the replication process. After a short transient phase the average fibre length remains constant during the exponential replication phase, but decreases steadily with time in the saturation regime. (b) 2D plot of the fibre length distribution during (blue) and after (red) the exponential replication process. During the saturation phase a distribution with a lower average fibre length develops compared to the replication phase, due to fibre growth being relatively slower than fibre breakage in the saturation regime with respect to the exponential replication regime. Furthermore, in the saturation regime, the fibre length distribution steadily varies over time, unlike what happens in the replication regime, where the constant fibre length distribution ensures exponential growth of the fibre population.

2.3.5 Model implementation details: systems of ODEs

The computational model was developed and implemented in C++. A solver for the set of ODEs (Ordinary Differential Equations) modelling the replicating system was coded explicitly at the programming language level in order to properly account for general mass balance and positive concentration constraints. At any point of the simulation, a “configuration” of the system is kept track of, i.e., a set of concentrations, and a new configuration for the current time step is computed by calculating incremental variations through numerical integration of the set of ODEs. The new configuration subsequently becomes the current one for the next time step. Other parameters are kept track of for more than one time step in order to ensure better approximation of the calculations. For example, the total hexamer

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concentration for three time steps is needed in order to calculate an O(t2₎

approximation (where t is the duration of the time step) of the replication rate for

the current time step. A configuration typically includes concentrations of monomers, linear dimers, cyclic trimers, tetramers, non-assembled hexamers and the concentration of fibres for every fibre length from a nucleus of 2 hexamers in a fibre up to a maximum length (i.e., number of hexamers in the same fibre). The code is tuneable on a number of parameters, among which the following: — Number of time steps that the simulation is run for

— Maximum possible length (in number of hexamers) of individual fibres — Monomer initial concentration

— Trimer initial concentration

— Non-assembled hexamer initial concentration

— Flag to keep the monomer concentration steady (constant along the simulation)

— Flag to keep the trimer concentration steady

— Flag to keep the non-assembled hexamer concentration steady

— Rate constants for the processes among the non-assembled species (described below)

— Rate constant for nucleation of hexamer fibres from non-assembled species — Breakage probability profile against fibre length

— Rate constant for sequestration of non-assembled hexamers on fibre ends — Rate constant for elongation of fibres from smaller macrocycles

— Flag to activate/deactivate fibre nucleation (if not active, artificial initial seeding is implemented)

— Flag to activate/deactivate fibre breakage

While the main function has the purpose to cycle through the time steps of the simulation, the function simulate integrates the system of ODEs, and the other functions serve different other fundamental roles, e.g., allocation/deallocation of memory for the simulation, and outputting/parsing of the results. The general purposes of the functions implemented in the software are as follows:

— simulate: integrates the system of ODEs and is therefore the core of the calculations;

— create: creates a new configuration by allocating memory for it; a configuration is a set of concentrations for solution library members (monomers, dimers, trimers, tetramers, hexamers) and replicating fibres up to a certain maximum fibre length;

— destroy: destroys a previously created configuration by deallocating its memory;

— cfgcopy: copies a configuration to another one;

— generatebp: defines a fibre breakage probability profile, where breakage probability increases linearly from zero to a maximum value between two predefined length thresholds;

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n — createpath: generates a path and filename based on the simulation parameters and outputs the result into a file called “lastpath.txt”;

— write: prints the results of the simulation into an output file; — parse: processes the entire output file and reformats numbers.

The system of ODEs is integrated within the simulate function. Much attention is dedicated to implementing a solver for this set of equations which respects mass balance and positive concentration constraints. The system includes the kinetic equations described for each individual process in the following (numbers are used to indicate the corresponding oligomers, i.e., 1 for monomers, 2 for dimers, …, 6na for

non-assembled hexamers, 6i for hexamers within fibres of length i, etc.):

−𝑑𝑑12_{𝑑𝑑𝑑𝑑 = 2}[1] 𝑑𝑑12_{𝑑𝑑𝑑𝑑 = 𝑘𝑘}[2] 12[1]2 −𝑑𝑑23_{𝑑𝑑𝑑𝑑 = −}[2] 𝑑𝑑23_{𝑑𝑑𝑑𝑑 =}[1] 𝑑𝑑23_{𝑑𝑑𝑑𝑑 = 𝑘𝑘}[3] 23[1][2] −𝑑𝑑24_{𝑑𝑑𝑑𝑑 = 2}[2] 𝑑𝑑24_{𝑑𝑑𝑑𝑑 = 𝑘𝑘}[4] 24[2]2 −𝑑𝑑36_{𝑑𝑑𝑑𝑑 = 2}[3] 𝑑𝑑36_{𝑑𝑑𝑑𝑑}[6𝑟𝑟𝑟𝑟]= 𝑘𝑘36[3]2 −𝑑𝑑46_{𝑑𝑑𝑑𝑑 = −}[4] 𝑑𝑑46_{𝑑𝑑𝑑𝑑 =}[2] 𝑑𝑑46_{𝑑𝑑𝑑𝑑}[6𝑟𝑟𝑟𝑟]= 𝑘𝑘46[2][4] −𝑑𝑑63_{𝑑𝑑𝑑𝑑}[6𝑟𝑟𝑟𝑟]=₂1𝑑𝑑63_{𝑑𝑑𝑑𝑑 = 𝑘𝑘}[3] 63[6𝑟𝑟𝑟𝑟] −𝑑𝑑64_{𝑑𝑑𝑑𝑑}[6𝑟𝑟𝑟𝑟]=₃2𝑑𝑑64_{𝑑𝑑𝑑𝑑 = 𝑘𝑘}[4] 64[6𝑟𝑟𝑟𝑟] −_{𝑖𝑖 − 1}1 𝑑𝑑𝑑𝑑𝑒𝑒𝑠𝑠𝑑𝑑𝑒𝑒𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟_{𝑑𝑑𝑑𝑑} [6𝑟𝑟−1]= −𝑑𝑑𝑑𝑑𝑒𝑒𝑠𝑠𝑑𝑑𝑒𝑒𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟_{𝑑𝑑𝑑𝑑} [6𝑟𝑟𝑟𝑟]=1_𝑖𝑖𝑑𝑑𝑑𝑑𝑒𝑒𝑠𝑠𝑑𝑑𝑒𝑒𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟_{𝑑𝑑𝑑𝑑} [6𝑟𝑟] = 𝑘𝑘𝑑𝑑𝑒𝑒𝑠𝑠𝑑𝑑𝑒𝑒𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟[6𝑟𝑟−1]_{𝑖𝑖 − 1}2 [6𝑟𝑟𝑟𝑟]

Where [6𝑟𝑟−1]_𝑟𝑟−12 is the concentration of fibre ends for fibres of length i-1 calculated from the corresponding concentration of hexamers which form fibres of such length. −_{𝑖𝑖 − 1}2 𝑑𝑑𝑒𝑒𝑟𝑟𝑟𝑟𝑟𝑟𝑒𝑒𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟_{𝑑𝑑𝑑𝑑} [6𝑟𝑟−1]= −𝑑𝑑𝑒𝑒𝑟𝑟𝑟𝑟𝑟𝑟𝑒𝑒𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟_{𝑑𝑑𝑑𝑑} [3]=2_𝑖𝑖𝑑𝑑𝑒𝑒𝑟𝑟𝑟𝑟𝑟𝑟𝑒𝑒𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟_{𝑑𝑑𝑑𝑑} [6𝑟𝑟] = 𝑘𝑘𝑒𝑒𝑟𝑟𝑟𝑟𝑟𝑟𝑒𝑒𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟[6𝑟𝑟−1]_{𝑖𝑖 − 1}2 [3]2 −𝑑𝑑𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑒𝑒𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟_{𝑑𝑑𝑑𝑑} [6𝑟𝑟𝑟𝑟]=𝑑𝑑𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑒𝑒𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟_{𝑑𝑑𝑑𝑑} [62]= 𝑘𝑘𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑒𝑒𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟[6𝑟𝑟𝑟𝑟]2 −𝑑𝑑𝑏𝑏𝑟𝑟𝑒𝑒𝑟𝑟𝑘𝑘𝑟𝑟𝑒𝑒𝑒𝑒_{𝑑𝑑𝑑𝑑} [6𝑟𝑟]= 𝑑𝑑𝑏𝑏𝑟𝑟𝑒𝑒𝑟𝑟𝑘𝑘𝑟𝑟𝑒𝑒𝑒𝑒�6𝑟𝑟 2� 𝑑𝑑𝑑𝑑 = 𝑘𝑘𝑏𝑏𝑟𝑟𝑒𝑒𝑟𝑟𝑘𝑘𝑟𝑟𝑒𝑒𝑒𝑒(𝑟𝑟)[6𝑟𝑟] −𝑑𝑑𝑏𝑏𝑟𝑟𝑒𝑒𝑟𝑟𝑘𝑘𝑟𝑟𝑒𝑒𝑒𝑒_{𝑑𝑑𝑑𝑑} [6𝑟𝑟]=_{𝑖𝑖 − 1}2𝑖𝑖 𝑑𝑑𝑏𝑏𝑟𝑟𝑒𝑒𝑟𝑟𝑘𝑘𝑟𝑟𝑒𝑒𝑒𝑒�6𝑟𝑟−1 2 � 𝑑𝑑𝑑𝑑 = 2𝑖𝑖 𝑖𝑖 + 1 𝑑𝑑𝑏𝑏𝑟𝑟𝑒𝑒𝑟𝑟𝑘𝑘𝑟𝑟𝑒𝑒𝑒𝑒�6𝑟𝑟−1 2 +1� 𝑑𝑑𝑑𝑑 = 𝑘𝑘𝑏𝑏𝑟𝑟𝑒𝑒𝑟𝑟𝑘𝑘𝑟𝑟𝑒𝑒𝑒𝑒(𝑟𝑟)[6𝑟𝑟]

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The latter two equations describe the breakage rates for fibres of even and odd length, respectively.

Hence, the full differential model for our system is the following (all possible fibre lengths i higher than 2 should be considered), where the technical detail about

fibres of odd length i being broken into fibres of length (i-1)/2 and (i-1)/2+1 has been

neglected for clarity (but is incorporated into the simulations):

−𝑑𝑑[1]_{𝑑𝑑𝑑𝑑 = 𝑘𝑘}12[1]2+ 𝑘𝑘23[1][2] −𝑑𝑑[2]_{𝑑𝑑𝑑𝑑 = −}1_{2 𝑘𝑘}12[1]2+ 𝑘𝑘23[1][2] + 𝑘𝑘24[2]2+ 𝑘𝑘46[2][4] −𝑑𝑑[3]_{𝑑𝑑𝑑𝑑 = −𝑘𝑘}23[1][2] + 𝑘𝑘36[3]2− 2𝑘𝑘63[6𝑟𝑟𝑟𝑟] + 𝑘𝑘𝑒𝑒𝑟𝑟𝑟𝑟𝑟𝑟𝑒𝑒𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟[6𝑟𝑟−1]_{𝑖𝑖 − 1}2 [3]2 −𝑑𝑑[4]_{𝑑𝑑𝑑𝑑 = −}1_{2 𝑘𝑘}24[2]2+ 𝑘𝑘46[2][4] −3_{2 𝑘𝑘}64[6𝑟𝑟𝑟𝑟] −𝑑𝑑[6_{𝑑𝑑𝑑𝑑 = −}𝑟𝑟𝑟𝑟] 1_{2 𝑘𝑘}36[3]2− 𝑘𝑘46[2][4] + 𝑘𝑘63[6𝑟𝑟𝑟𝑟] + 𝑘𝑘64[6𝑟𝑟𝑟𝑟] + 𝑘𝑘𝑑𝑑𝑒𝑒𝑠𝑠𝑑𝑑𝑒𝑒𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟[6𝑟𝑟−1]_{𝑖𝑖 − 1}2 [6𝑟𝑟𝑟𝑟] + 𝑘𝑘𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑒𝑒𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟[6𝑟𝑟𝑟𝑟]2 −𝑑𝑑[6_{𝑑𝑑𝑑𝑑 = −𝑘𝑘}2] 𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑒𝑒𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟[6𝑟𝑟𝑟𝑟]2 −𝑑𝑑[6_{𝑑𝑑𝑑𝑑 = −𝑖𝑖𝑘𝑘}𝑟𝑟] 𝑑𝑑𝑒𝑒𝑠𝑠𝑑𝑑𝑒𝑒𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟[6𝑟𝑟−1]_{𝑖𝑖 − 1}2 [6𝑟𝑟𝑟𝑟] + 𝑖𝑖𝑘𝑘𝑑𝑑𝑒𝑒𝑠𝑠𝑑𝑑𝑒𝑒𝑑𝑑𝑑𝑑𝑟𝑟𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟[6𝑟𝑟]2_𝑖𝑖[6𝑟𝑟𝑟𝑟] −_{2 𝑘𝑘}𝑖𝑖 𝑒𝑒𝑟𝑟𝑟𝑟𝑟𝑟𝑒𝑒𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟[6𝑟𝑟−1]_{𝑖𝑖 − 1}2 [3]2+_{2 𝑘𝑘}𝑖𝑖 𝑒𝑒𝑟𝑟𝑟𝑟𝑟𝑟𝑒𝑒𝑟𝑟𝑑𝑑𝑟𝑟𝑟𝑟𝑟𝑟[6𝑟𝑟]2_𝑖𝑖[3]2 + 𝑘𝑘𝑏𝑏𝑟𝑟𝑒𝑒𝑟𝑟𝑘𝑘𝑟𝑟𝑒𝑒𝑒𝑒(𝑟𝑟)[6𝑟𝑟] − 𝑘𝑘𝑏𝑏𝑟𝑟𝑒𝑒𝑟𝑟𝑘𝑘𝑟𝑟𝑒𝑒𝑒𝑒(2𝑟𝑟)[62𝑟𝑟] 2.4. CONCLUSIONS

We have demonstrated that replicators which, upon replication, assemble into large but fragile aggregates may be liberated by fracturing these aggregates through mild mechanical forces. Such fragmentation can overcome replicator self-inhibition and, most importantly, enables exponential replication, consistent with our experimental data and evident from numerical simulations. Notably, our simulations also indicate that, without breakage, the system reverts back to obeying the “square root law of autocatalysis”, albeit through a mechanism different from that postulated for duplex-forming replicators.

The growth/breakage mechanism may well provide a general solution to the auto-inhibition problem almost inherently associated with self-replication. We speculate that the replicators developed by Ashkenasy13 _{might well work by a similar} mechanism. Our mechanistic understanding gives clear guidance that may be employed to successfully design new exponential replicators, which opens up

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n realistic prospects of achieving Darwinian evolution in a purely synthetic chemical system. The next step in this area involves developing conditions in which replication and destruction operate in parallel, giving rise to a dynamic kinetic stability regime14 _{that is characteristic for life. Efforts in this direction are described} in Chapter 3 of this thesis.

2.5. EXPERIMENTAL METHODS

Library preparation and monitoring. Dynamic combinatorial libraries were prepared by dissolving building block 1, obtained from Cambridge Peptides, in a 50 mM pH 8.1 potassium borate buffer to a final concentration of 3.8 mM. The pH of the resulting solution was adjusted to 8.1-8.2 by addition of small amounts of a 2.0 M KOH solution. All libraries were contained in HPLC vials (12 × 32 mm) tightly closed with Teflon-lined snap caps. The libraries were stirred using a Teflon coated magnetic stirrer bar (5 × 2 mm, obtained from VWR), on an IKA RCT basic stirrer hotplate at 1200 rpm unless otherwise specified. Library compositions were monitored by quenching 2.0 µL samples of the library in 98 µL of a solution of doubly distilled H2O containing 0.6% TFA, in a glass UPLC vial, and injecting 5.0 µL of this sample on the UPLC. For samples that were monitored over time it was confirmed that the total peak area in the UPLC chromatograms remained constant. UPLC-MS analysis. UPLC analyses were performed on a Waters Acquity UPLC I-class system equipped with a PDA detector. All analyses were performed using a reversed-phase UPLC column (Aeris Widepore 3.6µm XB-C18 150 × 2.10 mm, purchased from Phenomenex). UV absorbance was monitored at 254 nm. Column temperature was kept at 35 °C. UPLC-MS was performed using a Waters Acquity UPLC H-class system coupled to a Waters Xevo-G2 TOF. The mass spectrometer was operated in the positive electrospray ionization mode. Injection volume was 5 µL of a 3.8 mM library subjected to a 1:50 dilution in a solution of 0.6 v% of trifluoroacetic acid in doubly distilled water. Eluent flow was 0.3 mL/min; eluent A: UPLC grade water (0.1 v% trifluoroacetic acid); eluent B: UPLC grade acetonitrile (0.1 v% trifluoroacetic acid). time (min) %A %B 0,0 90,0 10,0 1,0 90,0 10,0 1,3 75,0 25,0 3,0 72,0 28,0 11,0 69,0 31,0 11,5 5,0 95,0 12,0 5,0 95,0

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Table 2.5.1 | UPLC method. Eluent gradient used for UPLC analysis of libraries formed from building block 1 where A: UPLC grade water (0.1 v% trifluoroacetic acid); eluent B: UPLC grade acetonitrile (0.1 v% trifluoroacetic acid). Compound Retention time (min) m/z calculated m/z observed 1 3.8 760.35 [M+H]1+_, 380.68 [M+2H]2+ 760.32 [M+H]1+_, 380.65 [M+2H]2+ (1)2 6.2 1517.7 [M+H]+, 759.4 [M+2H]2+_, 506.6 [M+3H]3+_, 380.2 [M+5H]4+ 1517.5 [M+H]+_, 759.6 [M+2H]2+_, 506.8 [M+3H]3+_, 380.5 [M+5H]4+ (1)3 8.7 1137.50 [(M+1)+2H]2+, 759.01 [(M+2)+3H]3+_, 569.25 [(M+1)+4H]4+_, 455.61 [(M+1)+5H]5+ 1137.48 [(M+1)+2H]2+_, 758.98 [(M+2)+3H]3+_, 569.23 [(M+1)+4H]4+_, 455.58 [(M+1)+5H]5+ (1)4 6.8 1516.67 [(M+2)+2H]2+, 1011.45 [(M+2)+3H]3+_, 758.59 [(M+1)+4H]4+ 1516.64 [(M+2)+2H]2+_, 1011.42 [(M+2)+3H]3+_, 758.56 [(M+1)+4H]4+ (1)6 8.2 1516.67 [(M+3)+3H]3+, 1137.75 [(M+3)+4H]4+_, 910.80 [(M+5)+5H]5+_, 759.00 [(M+4)+6H]6+ 1516.62 [(M+3)+3H]3+_, 1137.72 [(M+3)+4H]4+_, 910.77 [(M+5)+5H]5+_, 758.99 [(M+4)+6H]6+ Table 2.5.2 | UPLC-MS compound identification.

Seeding experiments. A library was prepared by dissolving 3.8 mM 1 in 50 mM borate buffer at pH 8.2. The library was then oxidized up to 70% using a freshly prepared solution of sodium perborate (38 mM, pH 8.0). The composition of the mixture was at this point: 25% monomer 1, 5% linear dimer (1)2, 35% cyclic trimer (1)3 and 35% cyclic tetramer (1)4. The resulting solution was split into 4 samples of 200 µL and each one was then seeded with a pre-formed library rich in the hexamer of 1 (which had been continuously stirred at 1200 rpm), by adding respectively 5.0 mol%, 10 mol%, 15 mol% and 20 mol% of hexamer (the mol% was calculated as equivalents of 1 in the hexamer relative to equivalents of 1 in the library). The library was then stirred at 1200 rpm and the change in hexamer concentration was monitored by sampling every 8 min as described above.

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