What is special about autocatalysis?

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.

Graphical abstract

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 + X_j → 2X_j, (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, X_m defined by f_m = max{f₁, f₂,…, f_n}, 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,

$$ {\text{A}} + {\text{X}}\mathop{\longrightarrow}\limits_{}^{k}2{\kern 1pt} {\text{X}}, $$

(1)

is described by the kinetic equation and its solution curve:

$$ \frac{{{\text{d}}x}}{{{\text{d}}t}} = kax = - \frac{{{\text{d}}a}}{{{\text{d}}t}} = v{\kern 1pt} (t);\quad x(t) = \frac{{\left( {a{\kern 1pt} (0) + x{\kern 1pt} {\kern 1pt} (0)} \right)x{\kern 1pt} {\kern 1pt} (0)}}{{x{\kern 1pt} {\kern 1pt} (0) + a{\kern 1pt} {\kern 1pt} (0)e^{{ - {\kern 1pt} {\kern 1pt} k\left( {a{\kern 1pt} (0) + x{\kern 1pt} (0)} \right)t}} }}. $$

(1a)

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.

Fig. 1

Stochastic delay in the irreversible first-order autocatalytic reaction A + X → 2X. The figures compare expectation values (full thick lines), one standard deviation bands (thin light lines) and deterministic solutions (dashed thick lines) of the reactions A + B → 2C (upper part) and A + X → 2X (lower part). In the upper part, we see the results for the non-autocatalytic chemical reaction. The total number of particles is N = 100, the fluctuations in the order of $\sqrt N$, and the deterministic solution almost coincides with the stochastic expectation value E(C(t)). The shape of the solution curves is hyperbolic. The lower part presents the analogous curves for the autocatalytic reaction that are “S” shaped or sigmoid. The fluctuations are larger because of two effects: (1) thermal fluctuations are increased because of self-enhancement and (2) stochastic delay, which causes the expectation value E(X(t)) to be shifted to longer times. The deterministic curve is well separated from the expectation value (see text). Parameter choice: k = 0.01 [t⁻¹ C⁻¹], N = 100, A(0) = B(0) = 50, C(0) = 0 and A(0) = 99, X(0) = 1, respectively. [C] stands here for the unit of number density

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]:

$$ \frac{{{\text{d}}P_{M} }}{{{\text{d}}t}} = k{\kern 1pt} {\kern 1pt} \left( {(M + 1){\kern 1pt} (C - M - 1)P_{M + 1} - M(C - M)P_{M} } \right);\quad M,C \in {\mathbb{N}}. $$

(2)

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 P_M (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, P_M (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 P_M+1 and P_M, 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:

$$ {\text{A}} + {\text{B}}\mathop{\longrightarrow}\limits_{}^{k}2{\text{C}}. $$

(1c)

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.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: ΔX_max = max{x(t)  −  E(X(t))} = x(t_max) −E(X(t_max)) with t_max being the time, where the maximal difference occurs.² The difference ΔX_max can be converted into an incremental quantity δ that is approximately independent of the initial particle density X₀ = X(0) and population size N:

$$ \delta = \Delta {\kern 1pt} X_{{{ \rm max }\,}} \frac{{X_{0} }}{N}\;[{\text{C}}]\quad {\text{or}}\quad \Delta X_{\rm max } = \delta \frac{N}{{X_{0} }}\left[ {\text{C}} \right]. $$

(3)

Numerical values are shown in Table 1. The stochastic delay ΔX_max is proportional to the population size and inversely proportional to the initial number of autocatalyst molecules X. The stochastic delay ΔX_max vanishes with increasing X₀ and thus leads to convergence of the expectation value E(X(t_max)) 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 = ΔX_maxX(0)/N is insensitive to population size.

Table 1 Stochastic delay ΔX_max in the irreversible autocatalytic reaction ${\text{A}}\, + \,{\text{X}}\,\mathop{\longrightarrow}\limits_{}^{k}\,2{\text{X}}$

Introduction of the inverse reaction,

$$ {\text{A}} + {\text{X}} \mathop{\longleftarrow}\limits_{h}^{}2{\text{X}}, $$

(1′)

leads to little change in the reaction scenario. Instead of going into full turnover of A, the reaction converges to the thermodynamic equilibrium:

$$ \frac{{\left[ {\text{X}} \right]^{2} }}{{\left[ {\text{A}} \right]\left[ {\text{X}} \right]}} = \frac{{\left[ {\text{X}} \right]}}{{\left[ {\text{A}} \right]}} = \frac{k}{h} = K. $$

(4)

The stochastic delay ΔX_max 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:

$$ \frac{{{\text{d}}x}}{{{\text{d}}t}} = k(c{\kern 1pt} - x)x - hx^{2} = kcx - (k + h)x^{2} . $$

(1a′)

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.³ This yields the five-step reaction mechanism:

Fig. 2

The flow reactor as a device for studying autocatalytic reactions in an open system. The sketch shows a continuous-flow stirred-tank reactor (CFSTR). The interior of the reactor is assumed to be well mixed and at constant controllable temperature. A stock solution containing the resource A at concentration c₀ is assumed to flow into the reactor at a volume flow rate $r\; = \;V{\kern 1pt} {\kern 1pt} \cdot {\kern 1pt} {\kern 1pt} \tau_{\text{R}}^{ - 1}$ where V is the volume of the reactor and $\tau_{\text{R}}$ the mean residence time of a volume element in the reactor. The volume increase is exactly and instantaneously compensated by an outflow of the reaction mixture in the reactor

$$ *\mathop{\longrightarrow}\limits_{}^{{r \cdot c_{0} }}{\text{A}}, $$

(5a)

$$ {\text{A}} + {\text{X}}\mathop{\longrightarrow}\limits_{}^{k}2{\text{X}}, $$

(5b)

$$ 2{\text{X}}\mathop{\longrightarrow}\limits_{}^{h}{\text{A}} + {\text{X}}, $$

(5c)

$$ {\text{A}}\mathop{\longrightarrow}\limits_{}^{r}0, $$

(5d)

$$ X\;\mathop{\longrightarrow}\limits_{}^{r}\;0. $$

(5e)

The two chemical species in mechanism (5), A and X, are independent and hence the system is described by the differential equations:

$$ \frac{{{\text{d}}a}}{{{\text{d}}t}} = - kax + hx^{2} + (c_{0} - a)r\quad {\text{and}}\quad \frac{{{\text{d}}x}}{{{\text{d}}t}} = kax - hx^{2} - xr, $$

(6)

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):

$$ S_{0} :\;\;\bar{a}^{(0)} = c_{0} ,\;\bar{x}^{(0)} = 0\quad {\text{and}}\quad S_{1} :\;\;\bar{a}^{(1)} = \frac{{hc_{0} + r}}{k + h},\;\bar{x}^{(1)} = \frac{{k{\kern 1pt} c_{0} - r}}{k + h}. $$

(7)

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, lim_t→∞(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.

Fig. 3

The irreversible autocatalytic reaction A + X → 2X in a flow reactor of the type is shown in Fig. 2. The initial conditions are an empty reactor with seeding amounts of the autocatalyst: A(0) = 0 and X(0) = 1, 2, 3,…. The upper part of the figure shows the deterministic solution a(t) (black) and x(t) (red); the lower part presents a stochastic trajectory with A(t) (black) and X(t) (red). Four phases of the stochastic trajectory can be distinguished: (1) phase I: filling the reactor with the resource A, (2) phase II: decision to which (quasi)stationary state the trajectory is going to converge, (3) phase III: convergence towards the neighborhood of the (quasi)stationary state, and (4) phase IV: fluctuations around the (quasi)stationary state. In the deterministic trajectory phase II and phase III cannot be distinguished because of the uniqueness of solutions, and there are no fluctuations around the final state. Choice of parameters and initial conditions: N = 1000, k = 0.01 [t⁻¹ M⁻¹], r = 2.0 [t⁻¹ V⁻¹], A(0) = 0, X(0) = 1

Fig. 4

The course of deterministic and stochastic bifurcations in the reversible autocatalytic reaction A + X ↔ 2X in the flow reactor. In the deterministic process (upper part) uniqueness of trajectories implies the existence of basins of attraction, B(S): Trajectories from all points within the basin converge to the stationary state S. The autocatalytic reaction in the flow reactor has two basins, B(S₀) for the state of extinction and B(S₁) for the reaction state. The bifurcation occurs at time t = t_cr. The two basins are separated by a separatrix (red dotted line). In the stochastic approach deterministic trajectories are replaced by the probability distributions P(A) and P(X) (lower part). The distributions are monomodal before the bifurcation at time t = t_cr where they become bimodal. In the example sketched here, only A is present at the extinction state S₀ whereas both molecular species A and X are observed at the reaction state S₁

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).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.

Fig. 5

Two stochastic trajectories of the reversible autocatalytic reaction A + X ↔ 2X in the flow reactor with identical α-limits converging to two different states, extinction S₀ (upper plot) and reaction S₁ (lower plot). The two trajectories are computed with identical sets of parameters and initial conditions except different seeds s for the (pseudo)random number generator. Choice of parameters: N = 400, r = 0.5 [t⁻¹ V], k = 0.01 [t⁻¹ M⁻¹], h = 0.0005 [t⁻¹ M⁻²], seeds of the (pseudo)random number generator (Wolfram Mathematica, ExtendedCA): s = 491 (upper plot) and s = 919 (lower plot)

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).

Table 2 Frequency of convergence towards the two (quasi)stationary states of the autocatalytic reaction A + X ↔ 2X in the flow reactor

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 σ_A and σ_X 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:

$$ \frac{{{\text{d}}x}}{{{\text{d}}t}} = fx{\kern 1pt} \left( {1 - \frac{x}{C}} \right)\quad {\text{and}}\quad x(t) = \frac{x(0)C}{{x(0) + \left( {C - x(0)} \right)\exp ( - f{\kern 1pt} t)}}. $$

(8)

The parameter f is called the fitness of the species or subspecies⁵ 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:

$$ ({\text{A}}) + {\text{X}}\mathop{\longrightarrow}\limits_{}^{f}2{\text{X}}, $$

(9a)

$$ 2{\text{X}}\mathop{\longrightarrow}\limits_{}^{h}0. $$

(9b)

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, …, X_n}, results in a simple mathematical model for natural selection [16]. Concentrations and fitness values of subspecies are denoted by x = (x₁, x₂, …, x_n) and f = (f₁, f₂, …, f_n). With different fitness values for different subspecies, we obtain a Verhulst equation, which is generalized for heterogeneous populations:

$$ \frac{{{\text{d}}x_{j} }}{{{\text{d}}t}} = x_{j} \left( {f_{j} - \frac{{\sum\nolimits_{i = 1}^{n} {x_{i} } }}{C}\,\phi (t)} \right)\quad {\text{with}}\quad \phi (t) = \frac{1}{{\sum\nolimits_{i = 1}^{n} {x_{i} } }}\sum\limits_{i = 1}^{n} {f_{i} } {\kern 1pt} {\kern 1pt} x_{i} ;\quad j = 1, \ldots ,n. $$

(10a)

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:

$$ \frac{{{\text{d}}N}}{{{\text{d}}t}} = N\left( {1 - \frac{N}{C}} \right)\phi (t). $$

(10b)

This equation can be solved analytically and yields:

$$ N(t) = N(0)\frac{C}{{N(0) + \left( {C - N(0)} \right)\exp \left( { - \,\varPhi {\kern 1pt} (t)} \right)}}\quad {\text{with}}\quad \varPhi {\kern 1pt} (t) = \int_{0}^{t} {\phi {\kern 1pt} (\tau )\,{\text{d}}\tau } . $$

(10c)

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:

$$ \frac{{{\text{d}}\xi_{j} }}{{{\text{d}}t}} = \xi_{j} \left( {f_{j} - \phi {\kern 1pt} (t)} \right)\quad {\text{with}}\quad \phi {\kern 1pt} (t) = \sum\limits_{i = 1}^{n} {f_{i} } \,\xi_{i} ;\quad j = 1,2, \ldots ,n. $$

(10d)

Equation (10d) can be solved exactly by conventional techniques—for example, by integrating factor transformation—and finally we obtain:

$$ \xi_{j} {\kern 1pt} (t) = \frac{{\xi_{j} {\kern 1pt} (0)\,\exp {\kern 1pt} (f_{j} t)}}{{\sum\nolimits_{i = 1}^{n} {\xi_{i} {\kern 1pt} (0)\,\exp (f_{i} t)} }}. $$

(10e)

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 X_m with the largest fitness, f_m = max{f₁, f₂, …, f_n}:

$$ \sum\limits_{i = 1}^{n} {\xi_{i} {\kern 1pt} (0)\exp {\kern 1pt} \left( {f_{i} {\kern 1pt} {\kern 1pt} t} \right)} \approx \xi_{m} (0)\,\,\exp \left( {f_{m} \,t} \right)\quad {\text{for}}\;{\text{large}}\,\,t. $$

Insertion into Eq. (10e) leads to the long-time concentrations:

$$ \bar{\xi }_{j} = \frac{{\xi_{j} (0)}}{{\xi_{m} (0)}}\,\,\lim_{t \to \infty } \,\left\{ {\exp \left( {(f_{j} - f_{m} )t} \right)} \right\}, $$

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:

$$ \frac{d\phi }{dt} = \sum\limits_{i = 1}^{n} {f_{i\,} \dot{\xi }_{i} \; = \;\sum\limits_{i = 1}^{n} {f_{i} } } \,\left( {f_{{i{\kern 1pt} {\kern 1pt} }} \xi_{i} - \xi_{i} {\kern 1pt} {\kern 1pt} \sum\limits_{j = 1}^{n} {f_{j} {\kern 1pt} {\kern 1pt} \xi_{j} } } \right) = \left\langle {f^{2} } \right\rangle - \left\langle f \right\rangle^{2} = \text{var} \left\{ f \right\}. $$

(11)

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, X_j = 0 ∀ j = 1, 2,…, n), and n quasistationary states of selection, S_j = (A = r/f_j, X_j = C − r/f_j, X_i = 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 = X_m 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).

Table 3 Frequency of occurrence of (quasi)stationary states of selection between three species in the flow reactor

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]:

$$ {\text{A}} + {\text{X}}_{ + } \mathop{\longrightarrow}\limits_{}^{{k_{ - } }}{\text{X}}_{ - } + {\text{X}}_{ + } \quad {\text{and}}\quad {\text{A}} + {\text{X}}_{ - } \mathop{\longrightarrow}\limits_{}^{{k_{ + } }}{\text{X}}_{ + } + {\text{X}}_{ - } , $$

(12)

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].

Fig. 6

Drawn after Fig. 10 in [9]

Overall kinetics of complementary RNA replication by Qβ-replicase. The replication process in the batch reactor allows for a distinction of three different phases: (1) an exponential growth phase with excess enzyme, (2) a linear growth phase where practically all enzyme molecules are bound to RNA templates, and (3) a phase of saturation where very little or no RNA is produced because of product inhibition.

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:

$$ \begin{aligned} {\text{concentrations:}}\quad \left[ {{\text{A}}(t)} \right] = a(t), \, \left[ {{\text{X}}(t)} \right] = x(t), \ldots \hfill \\ {\text{number densities }}\left( {\text{particle numbers per unit volume}} \right){:}\left[ {{\text{A}}(t)} \right] = {\text{A}}(t), \ldots \hfill \\ \end{aligned} $$

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 ΔX_max at some time t_max.⁶ This quantity turns out to be proportional to the population size N and inversely proportional to the initial number of autocatalysts X(0):

Fig. 7

A sketch of an unambiguous computation of the stochastic delay in different stochastic models. The difference ΔX_max(t) = x(t) − E(X(t)) is scanned as a function of time and the maximum value of this difference, ΔX_max is determined

$$ \delta = \Delta X_{\rm max} \frac{N}{X(0)}. $$