Abstract
The single-step autocatalytic reaction A + X ↔ 2X in different environments—batch reactor, flow reactor, logistic equation—is studied by means of conventional deterministic kinetics and as a stochastic process. Self-enhancement requires an initial concentration of the autocatalyst X—at least in seeding amounts—for starting the reaction. Deterministic solution curves have sigmoid shapes. At small concentrations, three stochastic phenomena are observed: (1) thermal fluctuations, (2) stochastic delay, and (3) stochastic bifurcations and anomalous fluctuations in case of multiple final states. The introduction of heterogeneous populations containing subspecies with different fitness values gives rise to natural selection in all three environments investigated here. In large populations, survival of the fittest is observed, whereas random fluctuations may result in selection of each of the subspecies. Then, the fitness values determine only probabilities of selection. The fittest subspecies, of course, has the largest probability of selection. There is a smooth transition to neutral evolution where the probabilities of selection are the same for all subspecies.
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Introduction
The notion of autocatalysis has been introduced by the German chemist Wilhelm Ostwald [1] for the characterization of reactions that show an acceleration of the rate as a function of time. Autocatalytic reactions, however, were never popular in conventional chemistry and chemical technology mainly for two reasons: (1) autocatalysis gives rise to positive feedback, which is difficult to control and to handle in large reactors, and (2) autocatalytic systems may show complex dynamical phenomena such as bistability, oscillations, deterministic chaos, and spontaneous formation of spatial patterns or waves, which are not desirable in chemical production. The whole collection of these phenomena has been called nonlinear chemical dynamics by Irving Epstein and others [2, 3]. In the second half of the twentieth century, however, complex dynamics and its embedding in thermodynamics have been shifted into the center of physical interest. Experimental chemical reaction systems showing nonlinear phenomena, in particular oscillations and spatial pattern formation, were investigated [4, 5]. Autocatalysis was recognized as a basis for self-organization and has been extensively discussed in the context of models for the origin of life [6]. Oscillations in chemical reactions have been reported earlier [7] but these were singular studies and they took place in heterogeneous media. Beginning around 1960 with the studies of Boris Belousov and Anatol Zhabotinskii on a chemical reaction that exhibits oscillations in homogeneous solution, investigations on chemical self-organization became popular. Since then, a true wealth of experimental works and many theoretical studies on nonlinear chemical dynamics were performed [3].
A distinction between first- and higher order autocatalysis turned out to be very useful. First-order autocatalysis comprises all cases where a single autocatalytic particle is involved on the reactant side. The best and simplest example is the reaction A + X → 2X: a single autocatalyst particle X and a resource A produce together two particles X. This is the essence of asexual reproduction in biology, although X—being, for example, a virus particle or a bacterial cell—represents a highly complex entity and the resource A is not a single molecule but a variety of required building blocks. Simpler systems capable of reproduction such as RNA molecules have been studied by evolution in vitro [8, 9], but still reproduction is a complex multi-step process [10]. Nevertheless, single-step mechanisms describing the overall kinetics of the process under suitable conditions can be found (see also the section “Conclusion”). The most important phenomenon related to first-order autocatalysis is selection in the Darwinian sense: when several types of autocatalysts are present simultaneously all except one are eliminated through competition for resources.
Higher order autocatalysis is observed with several multi-step reactions often involving halogen—chlorine, bromine, and iodine—in different oxidation states. Investigations on higher order autocatalysis suffer even more than first-order autocatalytic systems from the fact that no experimental systems with simple one- or two-step mechanisms were found yet. Largely simplified but still realistic models such as the Oregonator involve five different elementary steps [11, 12]. A comparison of the individual steps in the Brusselator and the Oregonator models shows why the former is unrealistic: The Brusselator involves a termolecular step, A + 2X → 3X, which has extremely low probability because collisions of three particles occur very rarely. In the Oregonator, the termolecular step is resolved in several bimolecular steps. Oscillations in chemical reactions as well as deterministic chaos in chemistry have been studied in great detail (for an excellent review see, e.g., Ref. [3]). There is also extensive literature on Turing pattern formation [13] as well as spatiotemporal phenomena produced through nonlinear chemical dynamics [14].
Biology other than chemistry is centered around autocatalysis in the special form of reproduction. Multiplication of cells and organisms is in the core of biological thinking since it represents the basis of development and evolution. A second and not less important feature of biology—again in contrast to chemistry—is the necessity to deal with stochastic phenomena caused by small and very small particle numbers. Every mutant, after all, starts out from a single copy.
In this contribution, the focus is laid on first-order autocatalysis and its special role in chemical, biochemical and biological applications. At first, the reaction A + X → 2X is analyzed in the batch and in the flow reactors. Deterministic solutions are compared with the results of stochastic modeling at small particle numbers. The process A + X → 2X can be understood as a toy model for asexual reproduction. In biology, the logistic equation derived by Pierre-François Verhulst is often used for modeling reproduction and selection at the population level [15, 16]. We shall study it here as a simple alternative to the two classes of common chemical reactors. Broad emphasis is laid on the role of autocatalysis in natural selection. We shall study a population of independently reproducing variants A + Xj → 2Xj, (j = 1, 2,…, n), which are coupled only through the exploitation of a common resource A. The deterministic approach leads to selection of the fittest, Xm defined by fm = max{f1, f2,…, fn}, which is accompanied by optimization of fitness on the population level. The stochastic process occurring at small particle numbers results also in selection but apart from selection of the fittest, selection of less fit variants can happen as well. We emphasize: selection means only that all variants die out except the variant, which is selected. The stochastic selection process is guided by probabilities of selection that are determined by the fitness values.
Results and discussion
First-order autocatalysis in its simplest form, A + X → 2X, is studied under different environmental conditions: (1) the batch reactor, (2) the flow reactor, and (3) a ninetieth century model system for constrained growth, which is still in use and known as logistic equation. Autocatalysis leads to spectacular results when it operates on populations that contain collections of different autocatalysts, which live on the same resources: selection of one variant is observed. In the final part of the “Results and discussion” section, the relation between autocatalysis and Darwin’s natural selection is illustrated. Each section compares the deterministic solutions of conventional reaction kinetics with the results of the corresponding stochastic process that is modeled by a chemical master equation. The methods applied are described in the “Methods” section.
Autocatalysis in the batch reactor
The batch reactor [17] is a device that allows for studying chemical reactions in a closed system consisting here of well-mixed solutions with temperature control. The two conditions, spatial homogeneity and constant temperature, are commonly assumed to be fulfilled in the conventional theory of chemical reactions. Simply expressed, performing a reaction in the batch reactor implies that the reactants are perfectly mixed and then left alone in a thermostat.
The simple autocatalytic process,
is described by the kinetic equation and its solution curve:
The two variables a(t) and x(t) are constrained by the conservation relation a(t) + x(t) = c = const. The solution curve x(t) is “S” shaped or sigmoid. In reaction kinetics, this shape is typical for autocatalysis (Fig. 1). At low autocatalyst and large resource concentrations—x(t) « a(t)—the resource concentration remains approximately constant in the early phase of the reaction: a(t) ≈ a = const for small t, and increasing x(t) leads to self-enhancement since the reaction rate is proportional to the autocatalyst concentration: v(t) = k a x(t). An important feature of autocatalysis is the requirement of—at least—a seeding quantity of autocatalyst for the start of the reaction since the state of extinction S0 with x(0) = 0 implies no reaction, v(0) = 0 and ignition of the reaction does not occur.
Random fluctuations become important at small particle numbers when the discrete nature of molecular concentrations comes into play. The irreversible autocatalytic reaction, \( {\text{A}} + {\text{X}}\mathop{\longrightarrow}\limits_{}^{k}2{\text{X}} \), has been studied mathematically in form of the chemical master equation [18]:
This equation describes the time dependence of the probability distribution \( P_{M} (t) = {\text{Prob}}{\kern 1pt} \{ A(t) = M\} \) with the conservation relation C = A(t) + X(t). In the stochastic treatment, the probability PM (t) replaces the concentration variable a(t) and the second variable X(t) is defined by the conservation relation.
Although analytical expressions are available for the probabilities, PM (t) [18], they are of limited practical use because the results are obtained through rather sophisticated series that do not show universal convergence. In practice, a numerical approach based on sampling of trajectories (section “Methods”) is commonly preferred. Equation (2) has only one running index M and two terms PM+1 and PM, and computation as well as analysis of trajectories are fast and straightforward. The commonly presented stochastic results are expectation value \( E(A(t)) = \sum\nolimits_{M = 1}^{C - 1} {MP_{M} } (t) \) and standard deviation σ(A(t)), which are given in terms of one-standard-deviation error bands, E ± σ(A(t)). In Fig. 1, we compare a typical example of the autocatalytic process \( {\text{A}} + {\text{X}}\mathop{\longrightarrow}\limits_{}^{k}2{\text{X}} \) with a conventional, non-autocatalytic chemical reaction with similar stoichiometry:
In autocatalytic reactions, three types of random fluctuations can be distinguished: (1) thermal fluctuations, (2) stochastic delay, and (3) anomalous fluctuations caused by multiple (quasi)stationary states.Footnote 1 With irreversible reactions in the batch reactor, we observe only fluctuations of types (1) and (2), because we are dealing with a single stationary state only.
Stochastic delay is a newly reported stochastic phenomenon. The notion refers to the fact that the expectation value E(X(t)) appears delayed relative to the deterministic solution curve x(t) (Fig. 1, lower plot). As a quantitative measure for the delay, we consider the maximal difference between the curves: ΔXmax = max{x(t) − E(X(t))} = x(tmax) −E(X(tmax)) with tmax being the time, where the maximal difference occurs.Footnote 2 The difference ΔXmax can be converted into an incremental quantity δ that is approximately independent of the initial particle density X0 = X(0) and population size N:
Numerical values are shown in Table 1. The stochastic delay ΔXmax is proportional to the population size and inversely proportional to the initial number of autocatalyst molecules X. The stochastic delay ΔXmax vanishes with increasing X0 and thus leads to convergence of the expectation value E(X(tmax)) and the deterministic curve x(t). It is worth noticing that the stochastic delay increases with the total population size N within the range shown in Table 1, 100 ≤ N ≤ 1000, or in other words the relative stochastic delay δ = ΔXmaxX(0)/N is insensitive to population size.
Introduction of the inverse reaction,
leads to little change in the reaction scenario. Instead of going into full turnover of A, the reaction converges to the thermodynamic equilibrium:
The stochastic delay ΔXmax shows essentially the same behavior in the reversible and the irreversible cases.
Thermal fluctuations are present in all chemical reactions and their amplitudes are typically in the order of \( \sqrt N \), where N is the total number of molecules. Considering a single-step autocatalytic reaction with sharp initial conditions, \( P_{M} (0) = \delta_{{M,X_{0} }} \), the initial standard deviation is zero, σ(A(0)) = 0. It increases with time, passes through a maximum, decreases, and vanishes for the irreversible reaction. For the reversible reaction, it converges to its equilibrium value either after having passed a maximum or monotonously. The problem of extinction in the one-step autocatalytic reaction appears in different forms in the deterministic and the stochastic models. Because of the conservation relation a(t) + x(t) = c > 0, Eq. (1a) can be written in a single variable and we obtain for the reversible reaction:
The positive term dominates for small x, dx/dt > 0, x increases and hence the autocatalyst X cannot go extinct. In the stochastic system, the analysis gives a similar result for a different reason. The reaction step reducing the number of autocatalyst molecules, 2X → A + X, requires at least two molecules X, a single X molecule cannot undergo this conversion and hence X cannot die out. In other words, the state S1 = {(C-1)A, X} is a reflecting barrier and cannot be surmounted by the single-step autocatalytic reaction.
Autocatalysis in the flow reactor
A continuous-flow stirred-tank reactor (CFSTR) is sketched in Fig. 2. For a reaction in the reactor, inflow and outflow are modeled as pseudoreactions.Footnote 3 This yields the five-step reaction mechanism:
The two chemical species in mechanism (5), A and X, are independent and hence the system is described by the differential equations:
and sustains two stationary states: (1) the state of extinction S0 and (2) the reaction state S1. The values of variables at stationary states are readily calculated through equating to zero the two expressions in Eq. (6):
In the flow reactor, the conservation relation, a(t) + x(t) = c0, is not valid in general, but it is fulfilled at the steady states, \( \bar{a} + \bar{x} = c_{0} \), since the concentration in the reactor becomes equal to the concentration in the stock solution, limt→∞(a(t) + x(t)) = c0, for sufficiently long time. Stability analysis of the steady states by means of the eigenvalues of the Jacobian matrix is straightforward: the reaction state S1 is stable for volume flow rates r < k c0, whereas extinction occurs for larger flow rates. The stationary concentrations for the irreversible autocatalytic reaction are readily obtained by putting h = 0. Interestingly, the stationary concentrations at the steady states depend on the reaction parameter h but the eigenvalues of the Jacobian and the stabilities are the same in the irreversible and in the reversible cases.
Typical deterministic and stochastic trajectories converging to S1 for empty reactor initial conditions, no resource A (A(0) = 0) and the autocatalyst X in seeding quantities (X(0) = 1, 2, 3,…) are shown in Fig. 3. These initial conditions allow for the identification of different phases in the approach towards the long-time solutions and facilitate the analysis of anomalous fluctuations (Fig. 4). In phase I, the reactor is filled with resource A; the concentration A(t) increases until it is sufficiently large for making the reaction observable. Then, X(t) increases and A(t) decreases in phase II and approaches the long-time state in phase III. In phase IV, the system has eventually reached the steady state. The trajectories fluctuate around the expectation value corresponding to the error band E ± σ. For multiple long-time states in the stochastic system, a random decision in phase II determines the state towards which convergence occurs in phase III.
In case of multiple stationary states, the deterministic and the stochastic system behave differently. The uniqueness theorem for the solutions of differential equations requires that trajectories have well-defined α- and ω-limits within a single basin of attraction B(S).Footnote 4 Basins are separated by separatrices. Alternatively, we could state: deterministic trajectories do not cross. No uniqueness theorem of trajectories is valid for stochastic dynamics: arbitrarily many different trajectories may have identical α-limits and in most cases there are no simple ω-limits, for example, point limits. Instead, the trajectories are fluctuating around an expectation value and the ω-limit may be characterized best by an error band E ± σ. As we shall see in the section on “Selection”, there may be quasistationary states in the stochastic system that have no counterparts in the deterministic scenario.
Figure 4 sketches a deterministic and a stochastic bifurcation. Two features in which the stochastic approach differs from the deterministic solution are important here: (1) deterministic trajectories are unique and different trajectories cannot cross but stochastic trajectories can and (2) the deterministic trajectories have simple ω-limits, stochastic trajectories in general converge only in terms of the expectation value E, whereas the trajectory itself continues to fluctuate around E with a one-standard-deviation error band E ± σ. Deterministic trajectories are confined to one basin of attraction and stochastic trajectories are defined by probability distributions P that are extended over the entire domain of the stochastic variable. At a bifurcation point, deterministic trajectories coming from different basins of attraction separate, whereas the probability distribution of stochastic trajectories switches from monomodal to bimodal. Examples of stochastic trajectories converging to two different states, extinction S0 and reaction S1 are shown in Fig. 5.
Anomalous fluctuations [19] are the result of convergence of stochastic trajectories to alternative (quasi)stationary states. Then, the expectation value E calculated in the conventional way represents a weighted mean of the expectation values for the two quasistationary states covered by a bimodal distribution and accordingly, the standard deviation σ is very large. More illustrative than the broad distribution is the result of counting final states of stochastic trajectories (Table 2).
The results of trajectory counting are shown in Table 2 and reveal the expected dependence on initial conditions: the state of extinction S0 is most commonly approached for X(0) = 1 where it occurs with a frequency of about 78% for a population size of N = 100 that decreases to 28% when the population size is raised to N = 1000. At larger population sizes N, the frequency of extinction decreases faster with increasing number of initially present autocatalysts, X(0): for X(0) = 1 and N = 100 the percentage of extinctions lies around 78% and is reduced to 47% for X(0) = 3, whereas for N = 1000 it decreases from 27.9 to 1.3%. The standard deviations σA and σX illustrate the result of anomalous fluctuations in case of multiple (quasi)stationary states: the standard deviations become very large in cases of small initial values of X(0), which give rise to a high percentage of extinction.
Logistic equation
The logistic equation has been introduced in the first half of the nineteenth century by Pierre-François Verhulst as a quantitative model for exponential growth in systems with limited resources [20, 21]. He had been inspired by Thomas Robert Malthus who had discussed the disastrous consequences of uncontrolled exponential growth of human populations [22]. Malthus’ work has also been influential for the development of the concept of natural selection through Charles Darwin [23] and Alfred Russel Wallace [24]. Verhulst’s work was apparently forgotten and rediscovered several times in the early twentieth century [15, 25]. Despite its simplicity, Verhulst’s differential equation is still in use, for example, in population ecology for modeling microbial growth.
The logistic equation does not explicitly consider the resource A but limits the growth of a (homogeneous) population, \( \varPi \) = {X}, by means of a negative quadratic term:
The parameter f is called the fitness of the species or subspeciesFootnote 5 and the parameter C, the carrying capacity of the ecosystem, represents the maximal populations size N that is sustained stably by the environmental conditions. It is worth reconsidering the nature of the logistic approach to reproduction: The multiplication process is implicitly modeled by the ansatz dx/dt = f x, whereas growth limitation is represented in abstract mathematical form by the term − f x2/C. No hint is given on the physical nature of this limitation. It could be finite resources but also the consequence of overpopulation of the ecosystem without lack of nutrition.
In the deterministic approach, the logistic equation supports two stationary states, the extinction state S0 with \( \bar{x}^{(0)} = 0 \), and the saturation state S1 with \( \bar{x}^{(1)} = C \). At saturation, the population size adopts its maximal value, x = C. The logistic growth curve is sigmoid: It starts out like an exponential function and then after having passed an inflection point goes into saturation. Stability analysis is straightforward: S0 is unstable because small amounts of X introduced into the system lead to exponential growth in the early phase, S1 is asymptotically stable and will be approached from x-values above (x > C) and below (x < C) the carrying capacity.
The interpretation of logistic growth as a stochastic process is less straightforward. Here, we mention only one approach through modeling by means of a two-step chemical reaction, which gives to a rate equation with a positive linear term and a quadratic negative term as required for the limitation of growth. A negative quadratic term would also result, for example, from the reverse autocatalytic process, A + X ← 2 X, but as shown in the section on the “Batch Reactor” this mechanism would introduce a reflecting barrier at the state X = 1. Choosing an annihilation reaction for the introduction of the quadratic term turns out to be more suitable:
The expectation value embedded in the one standard deviation error band, E(X(t)) ± σ(X(t)), looks very similar to the lower part of Fig. 1; stochastic delay is observed and quantitatively the δ-value it is very close to the value calculated for the reaction in the batch reactor.
The stochastic model in Eq. (9) has two stationary states that correspond to S0 and S1 of the deterministic system (8). The state of extinction S0 is an absorbing barrier and the state of saturation S1 represents a quasistationary state.
Autocatalysis and natural selection
The replacement of the homogeneous population \( \varPi \)(t) = {X(t)} by a population that is heterogeneous with respect to the fitness values of subspecies, \( \varPi \)(t) = {X1, X2, …, Xn}, results in a simple mathematical model for natural selection [16]. Concentrations and fitness values of subspecies are denoted by x = (x1, x2, …, xn) and f = (f1, f2, …, fn). With different fitness values for different subspecies, we obtain a Verhulst equation, which is generalized for heterogeneous populations:
The function \( \phi \)(t) is the mean fitness of the population. We use \( N = \sum\nolimits_{i = 1}^{n} {{\kern 1pt} x_{i} } \) and reformulate the differential Eq. (10a) to get an equation for the growth of the total population:
This equation can be solved analytically and yields:
Equation (10c) is not autonomous because the distribution of subspecies and its time dependence are required for the calculation of \( \varPhi \)(t). The complete solution is derived easiest in terms of normalized concentrations for the subspecies, \( \xi \) = x/C with \( \sum\nolimits_{i = 1}^{n} {\xi_{i} } = 1 \), which fulfills the differential equation:
Equation (10d) can be solved exactly by conventional techniques—for example, by integrating factor transformation—and finally we obtain:
Insertion of (10e) into (10d) completes the solution.
Two properties of the solution of the extended Verhulst equation can be readily derived from Eq. (10e) and are of primary relevance for evolution: (1) selection of the fittest subspecies, and (2) optimization of the mean fitness of populations. Property (1) follows directly from Eq. (10e): we consider the non-neutral case—all f values are different—in the limit of long times. The sum in the denominator converges to the term containing the largest exponential and this is the term resulting from the subspecies Xm with the largest fitness, fm = max{f1, f2, …, fn}:
Insertion into Eq. (10e) leads to the long-time concentrations:
and provides the desired result: all \( \bar{\xi }_{j} \)-values are zero except \( \bar{\xi }_{m} = 1 \) and selection of the fittest has occurred. A proof for property (2) can be given straightforwardly through the calculation of the time derivative of the mean fitness:
Since a variance is always nonnegative, the mean fitness ϕ(t) is a non-decreasing function of time, it is optimized during the selection process and dϕ/dt vanishes if and only if \( \text{var} {\kern 1pt} {\kern 1pt} \{ f\} = 0 \), i.e., when all subspecies have the same fitness and the population has become homogeneous through the selection process.
The stochastic effects on selection are rather spectacular: instead of selection of the fittest every subspecies can be selected. Stochastic dynamics has one absorbing barrier corresponding to extinction of all subspecies, S0 = (A = C, Xj = 0 ∀ j = 1, 2,…, n), and n quasistationary states of selection, Sj = (A = r/fj, Xj = C − r/fj, Xi = 0 ∀ i, i ≠ j). The fitness values of the individual subspecies determine the probabilities of selection. A specific example of selection among three subspecies is shown in Table 3. Fitness values cover the entire range from neutrality, f1 = f2 = f3 = f, with equal probability of all three states S1, S2, and S3 to strong selection, where X1 = Xm is chosen preferentially. Under the initial condition of an empty reactor, the decision on the state towards which the trajectory converges is done by random fluctuations in phase II (Fig. 3).
It is worth noticing that natural selection and optimization of mean fitness occur in the population no matter whether one considers the logistic equation or autocatalysis in the batch or in the flow reactor. Remarkable indeed is the fact that during the nineteenth century no attempt was made of casting the principle of natural selection into rigorous mathematical form, although all mathematical tools including Verhulst’s equation have been available already 20 years before the publication of Darwin’s “Origin of Species”. The first true milestone of mathematics in evolution is Ronald Fisher’s book published as late as 1930 [26].
Conclusion
The first-order autocatalytic single-step reaction, A + X ↔ 2X, can be analyzed by mathematical tools in great detail and in diverse environments. Although pure single-step autocatalysis is extremely rare, the results derived here are of general importance since—as already mentioned in the introduction—complex many-step mechanisms may follow simple kinetics under certain conditions. As an example, we mention here the kinetics of plus–minus replication of RNA which was studied in great detail by Christoph Biebricher in the laboratory of Manfred Eigen [9, 10, 27]:
where X+ and X− are RNA molecules with complementary sequences. The growth curve of total RNA concentration in a batch reactor is sketched in Fig. 6. At low RNA concentration, perfect exponential growth as expected for the reaction \( {\text{A}} + {\text{X}}\mathop{\longrightarrow}\limits_{}^{k}2{\kern 1pt} {\text{X}} \) with \( k = \sqrt {k_{ + } {\kern 1pt} {\kern 1pt} k_{ - } } \) is recorded [27] and all properties of autocatalysis discussed in this contribution can be observed provided the conditions are chosen such that the reaction remains in the range of the exponential growth phase. A simplified mechanism of complementary replication has been proposed and analyzed [28].
In the high concentration limit of the deterministic approach autocatalysis shows two specific features that are absent in other reactions: (1) due to self-enhancement sigmoid solution curves are observed with single-step reactions and (2) autocatalytic reactions require (at least) seeding quantities of the autocatalyst for ignition of the reaction, because x(0) = 0 leads to x(t) = 0 \( \forall \)t—no reaction takes place.
Three stochastic features are observed with autocatalytic processes at low particle numbers: (1) thermal fluctuations that are the same as in every other chemical reaction and lead to \( \delta {\kern 1pt} {\kern 1pt} N\, \propto \,\sqrt N \) at thermodynamic equilibrium, (2) stochastic delay at small initial concentrations, and (3) anomalous fluctuations is case of multiple long-time states.
Stochastic delay with autocatalytic reactions to our knowledge has not been reported before in this form. It is a consequence of self-enhancement together with discretization at low particle numbers. Interestingly, the dependence of stochastic delay on initial particle numbers and population size is rather simple and an increment δ has been derived that is fairly constant and independent of the parameters mentioned.
Anomalous fluctuations are dominant at very small initial particle numbers and result from stochastic bifurcations in case of multiple (quasi)stationary states. Such small particle numbers are commonly not important in chemistry but they play an important role in evolution. Every mutant after all starts out from a single copy.
Methods
All numerical computations reported here were performed with Wolfram Mathematica, versions 7-11. Deterministic solution curves of kinetic differential equations were calculated with the Mathematica ODE integration routine.
All quantities with physical dimensions are given in arbitrary units, which are written in square brackets. This implies that in comparison with experimental data the appropriate unit element can be inserted. For example, time can be expressed as [t] = min, sec, msec, μs, etc. The unit elements used are time [t], volume [V], concentration [M] = [mol V−1], and number density [C] = [particles V−1]. Molecules are denoted by upper case letters: A, X, etc. The notation used for the most frequently used variables is:
Stochastic trajectories were computed by the stochastic simulation algorithm (SSA) developed by Daniel Gillespie (for reviews see Refs. [29, 30]). We used here a biologically motivated simulation routine for Gillespie’s algorithm, which has been developed by Bruce Shapiro within the xCellerator software design project [31].
The standard deviation is derived from the second moment of the distribution and, therefore, more sensitive to random fluctuations than the first moment, the expectation value. Accordingly, the corresponding quantities in Tables 2 and 3 are less accurate.
Calculation of the increment δ of the stochastic delay: the calculation of the stochastic delay is sketched in Fig. 7. The quantity that is extracted from the deterministic solution curve x(t) and the stochastic expectation value E(X(t)) is the maximum difference ΔXmax at some time tmax.Footnote 6 This quantity turns out to be proportional to the population size N and inversely proportional to the initial number of autocatalysts X(0):
Notes
In stochastic dynamical systems, it is appropriate to distinguish absorbing and quasistationary states. When a system reaches an absorbing state, it will stay there forever. A quasistationary state behaves like a true stationary or absorbing state for all practical purposes but a gigantic fluctuation of extremely low probability may transfer the system into an absorbing state.
It might seem more natural to define the stochastic delay as the maximal time difference between the two curves, ∆tmax = max{t|E(X(t)) = γ − t|x(t) = γ}. The results of this calculation are qualitatively the same as those obtained with ∆Xmax.
α- and ω-limits characterize the beginning and the end of a(n infinite) trajectory.
Fitness, in general, counts the number of fertile offspring that carry the genetic information into the next generation. A subspecies is a variant within a species consisting of individuals with the same fitness.
It could seem more natural to use the maximum difference along the time axis, Δtmax for the calculation of the stochastic delay. The qualitative results obtained are the same as those shown in Table 1 but the numerical computations are more involved.
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Acknowledgements
Open access funding provided by University of Vienna. Support of the work by the University of Vienna and the Santa Fe Institute is gratefully acknowledged. The author acknowledges many fruitful discussions with Professors Ivo L. Hofacker and Christoph Flamm. The public accessibility of the stochastic simulation algorithm (SSA) within the xCellerator project by Bruce Shapiro is acknowledged.
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Dedicated to Heinz Falk on the occasion of the 80th anniversary of his birthday.
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Schuster, P. What is special about autocatalysis?. Monatsh Chem 150, 763–775 (2019). https://doi.org/10.1007/s00706-019-02437-z
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DOI: https://doi.org/10.1007/s00706-019-02437-z