Chaos, Solitons and Fractals 35 (2008) 313–319
www.elsevier.com/locate/chaos
Error threshold ghosts in a simple hypercycle with error
prone self-replication
Josep Sardanyés
*
ICREA-Complex Systems Lab, Universitat Pompeu Fabra (GRIB), Dr Aiguader 80, 08003 Barcelona, Spain
Accepted 14 May 2006
Abstract
A delayed transition because of mutation processes is shown to happen in a simple hypercycle composed by two
indistinguishable molecular species with error prone self-replication. The appearance of a ghost near the hypercycle
error threshold causes a delay in the extinction and thus in the loss of information of the mutually catalytic replicators,
in a kind of information memory. The extinction time, s, scales near bifurcation threshold according to the universal
square-root scaling law i.e. s (Qhc Q)1/2, typical of dynamical systems close to a saddle-node bifurcation. Here,
Qhc represents the bifurcation point named hypercycle error threshold, involved in the change among the asymptotic
stability phase and the so-called Random Replication State (RRS) of the hypercycle; and the parameter Q is the replication quality factor. The ghost involves a longer transient towards extinction once the saddle-node bifurcation has
occurred, being extremely long near the bifurcation threshold. The role of this dynamical effect is expected to be relevant in fluctuating environments. Such a phenomenon should also be found in larger hypercycles when considering the
hypercycle species in competition with their error tail. The implications of the ghost in the survival and evolution of
error prone self-replicating molecules with hypercyclic organization are discussed.
Ó 2006 Elsevier Ltd. All rights reserved.
1. Introduction
One of the key questions in the appearance and evolution of the first living-like systems is how genetic information
was stored and transmitted generation to generation by the first replicator molecules as nucleic acids or their analogs
[1–8]. The well-known information catastrophe of replicator populations with quasi-species distribution might have
supposed an important evolutionary restriction for the maintenance of a stable genetic message coded by earlier replicators [1,2,4,8,9]. It is known that the length of earlier, error prone self-replicating systems is limited by the accuracy of
replication, and thus also the stability of a large genetic message [2,9,10]. Therefore, the existence of a critical mutation
rate beyond which genetic information becomes random might have not allowed the possibility to maintain a large
information message able to evolve under Darwinian selection. This catastrophic phenomenon, named error threshold
transition, has been analyzed and discussed in many studies [2,6,8,9,11–14]. It is known that this transition is sharp and
governed by a first-order phase transition [6,13,15,16] and can actually have relevant implications in RNA viruses
*
Corresponding author. Tel.: +34 935422834; fax: +34 932213237.
E-mail address: [email protected]
0960-0779/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved.
doi:10.1016/j.chaos.2006.05.020
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J. Sardanyés / Chaos, Solitons and Fractals 35 (2008) 313–319
dynamics and pathology [11–14]. Earliest replicators would have evolved under this informational restriction because
such molecules would not have presented complex enzymatic machineries able to ensure an error free self-replication.
Hence, as stated by the so-called Eigen’s paradox [3]: it is neither possible complex genomes without complex enzymatic
machineries nor complex enzymatic machineries without complex genomes.
The hypercycle hypothesis [1,2] states that a larger genetic message can be achieved with the integration of a set of
physically-unlinked informational carriers. Therefore, the system as a whole, in a kind of division of labour, can overcome the information threshold because the length of each of the species forming the hypercycle maintains below the
critical chain size. The integration of all the self-replicating species is produced by means of symbiotic processes given
by the catalytic coupling among the hypercycle species, which provides stable and controlled coexistence of all hypercyclic macromolecules [2]. In such a catalytic system, each species catalyzes the self-replication of the next one in a cyclic
architecture, forming a structure of greater complexity, which behaves as a single multimolecular coherent entity
[1,2,5,7,8,17]. The study of the hypercycle is extremely interesting because it is not only restricted to the theoretical analyses of the dynamical behavior of earlier replicator molecules in the context of the origin of life but can also be applied
to characterize present-day systems. For instance, the hypercyclic coupling has been reported in a short infection phase
of the coliphage Qb, where the RNA and viral protein grow hyperbolically in an autocatalytic loop [18]. The hypercycle
model can also be used to study symbiotic processes between different species [19]. Moreover, the hypercycle cooperation among self-replicating species and the crossed enhancement in reproduction has been experimentally shown with
artificial peptides [17]. It is worth to mention that hypercycle dynamics might also be the basis to analyze autocatalytic
as well as cross-catalytic chemical kinetics. From an evolutionary point of view, as previously mentioned, the hypercycle
has been suggested as a fundamental step to increase genome size in earlier living-like chemical systems [1,2,4,5,7,8]. In
this context, a phase of nonlinear evolution might have characterized an important step before the appearance of the
first living cell [2], in which heredity in terms of chemistry could have evolved on surfaces [20], where dispersal is reduced
and cooperation likely to happen [3].
The saddle-node bifurcation represents the basic mechanism for the creation and destruction of equilibrium points
[22]. When two fixed points collide annihilating each other at the bifurcation, it is possible to see the appearance of a
saddle remnant in phase space able to continue attracting the flows. This phenomenon, which is caused by the so-called
saddle ghost [22], involves a delayed transition because trajectories pass through a bottleneck region before another
fixed point is reached. Delayed transitions have been found in some physical systems [23–27] and in neural dynamics
[28,29].
Ghost-induced delayed transitions have been described in the field of condensed matter physics, specifically in models of charge density waves [27]. Moreover this phenomenon can also be found in autocatalytic and cross-catalytic selfreplicating systems. In this context, the hypercycle is able to delay its extinction by means of the ghost, which appears
once a critical degradation rate is overcome [30–32]. It is well known that the hypercycle is also sensitive to mutation
processes, when the accuracy of replication of the hypercycle members is considered. The error propagation in hypercycles has been studied [21,33]. For instance, the hypercycle composed by two indistinguishable species with error prone
self-replication is shown to get extinct when the mutation rate is very large, and thus the so-called hypercycle error
threshold is overcome [21]. This work is focused in this phenomenon. Here the extinction dynamics and the consequent
loss of information for this simple hypercycle competing with the so-called error tail is explored. The hypercycle error
threshold involves a saddle-node bifurcation that also causes the ‘‘apparition’’ of a ghost. The time delay behavior near
bifurcation point as well as the scaling properties of this ‘‘genetic’’ ghost are studied.
2. Deterministic model
The deterministic model described by Nuño and co-workers [21] is used to characterize the temporal evolution of a
system where a hypercycle formed by two kinetically indistinguishable species, I1 and I2, competes with their mutant
species. Such mutant replicators, which might correspond to a kind of hypercycle parasite, are also considered indistinguishable and are grouped in the so-called error-tail, I0 (see Fig. 1 left). The mathematical model for this system
is given by the next three coupled ODEs [21]:
x_ 1 ¼ ½QðA þ kx2 Þ /x1 ;
x_ 2 ¼ ½QðA þ kx1 Þ /x2 ;
x_ 0 ¼ ½A0 /x0 þ ð1 QÞ½Aðx1 þ x2 Þ þ 2kx1 x2 :
ð1Þ
ð2Þ
ð3Þ
Here the concentration variables xi(i = 0, 1, 2) = Xi/N, where Xi denotes the number of molecules of species Ii, and N
corresponds to the total amount of molecules in the system. A and A0 are the so-called amplification factors, which
J. Sardanyés / Chaos, Solitons and Fractals 35 (2008) 313–319
1
AQ
I1
k
I0
k
A0
+
AQ
1–Q
1
(B)
0.8
+
I2
1
(A)
1–Q
315
(C)
0.8
0.6
1st bifurcation
0.8
0.6
2nd bifurcation
0.6
0.4
0.4
0.4
0.2
0.2
0.2
0
0
0.2
0.4
0.6
0.8
1
1
0
0
0.2
0.4
0.6
0.8
Quality factor, Q
1
0
0
0.2
0.4
0.6
0.8
1
0
Fig. 1. Left: schematic graph structure of the hypercycle composed by two kinetically indistinguishable species, I1, 2, with error prone
self-replication. The system includes the hypercycle mutants in the so-called error tail compartment, I0. Here circle arrows represent the
non-catalyzed self-replication reactions, and the arrows connecting both hypercycle species indicate the catalytically assisted selfreplication given by the neighbor template. Right: (x1,x2) phase portraits i.e. simplex w2, showing the three possible scenarios of the
catalytic network analyzed by Nuño and co-workers [21]. At decreasing values of the quality factor, Q (represented with the horizontal
dashed line below phase space), the dynamical system undergoes two saddle-node bifurcations. In (A) Q = 0.95 > Qc with A = 1,
A0 = 0.75 and k = 1. In (B) Qhc < Q = 0.88 < Qc and (C) Q = 0.8748 < Qhc, both with A = 1, A0 = 0.95 and k = 1. Note that the
second bifurcation implies the asymptotic extinction of the hypercycle. However, the flows continue to be attracted by the region where
the saddle and the coexistence node coalesce. The arrows in the trajectories indicate the direction of the flows.
actually represent, respectively, the malthusian growth i.e. linear chemical formation, of hypercycle species and of the
error tail. A relevant parameter for the long-term dynamical behavior of Eqs. (1)–(3) is given by the quality factor, Q.
Thus being (1 Q) the mutation rate for the hypercycle species. Finally, k is the cross-catalytic self-replication parameter and / is the outflow (which maintains the total population constant i.e. CP constraint), with:
/ ¼ Aðx1 þ x2 Þ þ A0 x0 þ 2kx1 x2 ;
ð4Þ
thus being x1 and x2 independent variables. The trajectories of this dynamical system are embedded in a two-dimensional subspace, w2, the so-called simplex of the system, defined as w2 ¼ f~
x 2 R3 jx0 þ x1 þ x2 ¼ 1g. The dependence
of the equilibrium points of Eqs. (1)–(3) on the replication quality factor, Q, involves the appearance of two bifurcation
points [21], named Qc and Qhc, and given by:
A0
;
A
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
Qhc ¼ 2 f4kðA þ A0 Þ þ 32k 2 A0 ð2A þ kÞg:
2k
Qc ¼
ð5Þ
ð6Þ
The bifurcation point (6) is lower than the quasi-species one, Qc, because of the hypercyclic organization [34]. The
stability properties of the equilibrium points of Eqs. (1)–(3) have been completely analyzed (see [21,35,36]). This system
can display three different and well-defined scenarios according to the quality factor, Q, which are represented in Fig. 1.
ð1Þ ð2Þ
The first one corresponds to the case Q > Qc, where the only asymptotic stable state is the fixed point ðxþ ; xþ Þ (the
equilibrium point (v) in [21]). Here the hypercycle extinction point ðx1 ¼ 0; x2 ¼ 0Þ (the equilibrium point (i) in [21]) is
unstable and thus both hypercycle members asymptotically coexist (see Fig. 1A). When Q decreases below Qc (thus
mutation rate increases) but is maintained above Qhc, a first bifurcation takes place. This saddle-node bifurcation forces
ð2Þ
a scenario of dynamical bistability because of the appearance of the non-stable fixed point i.e. saddle point, ðxð1Þ
; x Þ
(the equilibrium point (iv) in [21]), which makes trajectories to flow towards the extinction attractor (which is now stað1Þ ð2Þ
ble), or towards the coexistence invariant point ðxþ ; xþ Þ, dependending on the initial condition of the hypercycle
replicators (see Fig. 1B). Finally, when Q < Qhc, the saddle and the coexistence nodes collide undergoing another saddle-node bifurcation in which the stable extinction point becomes the only attractor of the simplex, thus being globally
stable. This case corresponds to the so-called Random Replication State (RRS) (see [21] for details), where the mutation
rate for both hypercycle replicators is so high that the hypercyclic network is not able to maintain itself and thus information is lost (see Fig. 1C).
Lets numerically study this last scenario, thus focusing on the extinction dynamics near the bifurcation threshold
when considering the RRS, by means of the standard fourth-order Runge–Kutta method (the time stepsize used in this
work is dt = 0.1). The extinction time evolution of the hypercycle members within the RRS is represented in Fig. 2. Here
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J. Sardanyés / Chaos, Solitons and Fractals 35 (2008) 313–319
0.1
0.075
bottleneck
x
0.05
0.025
0
2000
4000
6000
8000
time
Fig. 2. Delayed extinction collapse for the hypercycle species I1, 2, indicated with x x1 = x2, for five decreasing values of the quality
factor, Q, all of them below the hypercycle error threshold, Qhc. In all the analyses of this work extinction is assumed with a replicator
concentration xi 6 108. Here, the five trajectories are numerically obtained with A = 1, A0 = 0.75, k = 1, and from right to left:
Q = 0.74262, Q = 0.742615, Q = 0.74261, Q = 0.7426 and Q = 0.7425.
10 5
τ
β = –1/2
10
4
10 -8
10 -4
10 -6
φ
Fig. 3. Square-root scaling law near bifurcation threshold for the symmetric two-member hypercycle with error prone self-replication.
Here the extinction time, s, is shown as a function of / = (Qhc Q) on a doubly logarithmic scale. As / ! 0, the quality factor, Q,
approaches from below, to the bifurcation point, Qhc. Parameter values are the same as in Fig. 2, with Qhc 0.74264.
the quality factor is below the hypercycle error threshold, thus the system has crossed the critical mutation point (this
case corresponds to the scenario C described in Fig. 1). Specifically, Fig. 2 shows five trajectories for five different values
of the quality factor. It is clearly shown that trajectories do not quickly collapse to extinction. On the contrary, they
undergo the characteristic plateau given by the bottleneck region of the ghost, which causes the slow passage of the
flow. Such a figure also shows that as the quality factor is decreased below Qhc i.e. mutation rate increases, the trajectories decrease the time spent in the ghost. The time delay dependence on the bifurcation parameter near the bifurcation
threshold typically follows a square-root scaling [22]. Such a scaling law has been described and discussed in other
dynamical systems [22,30–32]. Fig. 3 shows that the ghost for the system analyzed here also follows the square-root
scaling law. It actually means that once the quality factor is below the critical Qhc value, althoug the system asymptotically becomes extinct, the hypercycle is able to maintain the information during a longer time before collapsing.
3. Discussion
In the present work, the dynamics of information collapse in a hypercycle composed by two indistinguishable selfreplicating species with error prone self-replication is analyzed by using the model of Nuño and co-workers [21]. This
J. Sardanyés / Chaos, Solitons and Fractals 35 (2008) 313–319
317
model predicts a saddle-node bifurcation when the hypercycle error threshold is overcome i.e. large mutation rates.
After this bifurcation, the extinction point is the only invariant state of the dynamical system. However, extinction is
not quickly achieved near bifurcation threshold because of the appearance of a ghost in phase space. This ghost
makes trajectories to undergo a slow passage through a bottleneck region, responsible of delaying the flow before
extinction. Moreover, consistently with other dynamical systems [22,27,30,32], the extinction time, s, scales according
to s / = (Qhc Q)1/2 when / ! 0, thus indicating the universal nature of the square-root scaling law near the
bifurcation point. The presence of this phenomenon in the symmetric, two-component network analyzed in this paper
might also indicate the probable presence of the error threshold ghost in larger hypercycles competing with their
error tail.
Hypercycle delayed extinctions have been interpreted as a way to force a concentration memory in phase space
able to recover the system in fluctuating environments [30–32]. Hence, when considering the extinction delayed transition because of the hypercycle error threshold, the ability of delaying the flow could also suppose a selective advantage in more realistic, fluctuating environments, where the hypercycle should be able to recover and thus maintain the
information. The 2-fold feedback of the hypercycle organization has been shown to happen in a short phase of the
infection cycle of the RNA bacteriphage Qb [18]. In this context, the delayed extinction given by the ghost could
ensure the resilience of the viral particles inside the host beyond the bifurcation, therefore increasing their chances
of survival and infection successful within the host cell. As shown in this work, the extinction because of an increase
in mutation rate, as well as because of an increase in the degradation parameter (or equivalently, a decrease in the
self-replication rate) [30–32] would involve, considering the hypercycle organization, the bifurcation scenario and the
appearance of the ghost with all its dynamical implications. The genomes of even the smallest viruses encode several
functions, being replication the most important one [6,13]. In this sense, and as demonstrated by Eigen and co-workers [18], the possibility of the presence of a hypercycle coupling in the quasi-species populations of RNA viruses,
could have relevant importance in the life cycle and associated pathology of such viruses. The design of antiviral
strategies based on virus entry into error catastrophe have been proposed [13], and have been recently reviewed
[14]. Combined or synergic treatments with mutagens as well as with antiviral inhibitors might decrease viral attack,
although it is not clear which might be the better strategy [14], considering that different types of viruses might have
different reponses to mutagens and to antiviral inhibitors, thus having different infectious properties and kinetics [13].
The possible presence of autocatalytic or cross-catalytic processes in viral growth opens the possibility for the appearance of delayed extinction phenomena. In this context the entry into error catastrophe with mutagens as well as the
extinction of viral genomes because of a decrease in self-replication could be slowed down i.e. delayed, because of
the ghost, becoming an obstacle in the elimination of virus during infectious processes. The dynamical properties
of the ghost could also suppose a handicap for the immune system in facing hypothetic viral growth phases with
hyperbolic dynamics i.e. hypercyclic coupling. Hence, the effect of the ghost should be considered in the persistence
and infection of RNA viruses with 2-fold kinetic feedbacks.
It has also been discussed the possible role of the ghost in noisy media [32]. The interest in the qualitative theory of
stochastic dynamical systems has managed to analyze the hypercycle explored in this work from a more physically realistic point of view, considering random external fluctuations in the quality factor. In this context, Nuño and co-workers
[21] showed that when external noise is considered, the region of asymptotic stability for the hypercycle is decreased.
According to this result, extinction would be more probable when considering noisy i.e. random perturbations. The
stochastic analysis of the error threshold in a hypercycle has shown that the only asymptotic state of this system is
an absorbing boundary i.e. extinction [34]. However, this state has been considered irrelevant because the actual time
to reach this stationary state may be astronomically long [34]. The results of this work suggest that the ghost could be
the origin of such extremely long extinction transients. Moreover, the deterministic approach also suggests that if the
range of variation in the noisy quality factor fluctuates near the error threshold bifurcation point, the hypercycle would
perhaps recover because the saddle point would be continuously created and destroyed, and the hypercycle trajectories
could flow again towards the coexistence point because of the information memory due to the ghost, therefore avoiding
extinction and thus being able to self-maintain the information of the entire network. It has been pointed out that the
effect of noise in asymmetric hypercycles would be more dramatic and that the extinction level would play a more
important role in the survival of the hypercyclic network [21,36]. In this sense, and related to the delayed transition,
deterministic models have shown that the asymmetry of the hypercycle can prolong the extinction time by means of
the ghost as opposed to the symmetric case [32]. Thus, this effect could also be found when considering asymmetric
hypercycles with mutation processes and noise.
The role of noise in hypercycle dynamics has been explored in some works [21,35]. In this sense, not only the effect of
external noise (influence of the environment) should be considered, but also the internal noise (finite size effects) inherent to chemical reactions [34,37,38]. These stochastic approaches related to hypercycle delayed extinctions will be
explored in future research.
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J. Sardanyés / Chaos, Solitons and Fractals 35 (2008) 313–319
Acknowledgements
The author thanks Ricard V. Solé for useful and illuminating discussions. This work has been supported by an EU
PACE grant within the 6th Framework Program under contract FP6-002035 (Programmable Artificial Cell Evolution).
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