Autocatalytic Replication in aCSTR and ConstantOrganizationRobert HappelPeter F. Stadler
SFI WORKING PAPER: 1995-07-062
SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent theviews of the Santa Fe Institute. We accept papers intended for publication in peer-reviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our externalfaculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, orfunded by an SFI grant.©NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensuretimely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rightstherein are maintained by the author(s). It is understood that all persons copying this information willadhere to the terms and constraints invoked by each author's copyright. These works may be reposted onlywith the explicit permission of the copyright holder.www.santafe.edu
SANTA FE INSTITUTE
Autocatalytic Networks with Translation
Robert Happela� Robert Hechta� and Peter F� Stadlera�b��
aTheoretische Biochemie� Institut f�ur Theoretische Chemie
Universit�at Wien� Vienna� Austria
bSanta Fe Institute� Santa Fe� NM
�Mailing Address�Peter F Stadler� Inst� f� Theoretische Chemie� Univ� Wien
W�ahringerstr� ��� A�� Wien� AustriaPhone� �� � � ��� ��� Fax� �� � � ��� ��
E�Mail� studla�tbi�univie�ac�at
Happel et al�� Autocatalytic Networks with Translation
Abstract
We consider the kinetics of an autocatalytic reaction network in which replication and catalytic action is separatedby
a translation step� We �nd that the behavior of such a system is closely related to second order replicator equations�
which describe the kinetic of autocatalytic reaction networks in which the replicators act also as catalysts� In fact�
the qualitative dynamics seems to be described almost entirely be the second order reaction rates of the replication
step� For two species we recover the qualitative dynamics of the replicator equations� Larger networks show some
deviations� however� A hypercyclic system consisting of three interacting species can converge towards a stable
limit cycle in contrast to the replicator equation case� A singular perturbation analysis shows that the replication
translation system reduces to a second order replicator equation if translation is fast� The in�uence of mutations
on replication translation networks is also very similar to the behavior of selection�mutation equations�
�� Introduction
A variety of self�replicating chemical systems have been constructed and investigated experimen�
tally in the past �� years since Spiegelman�s �� in vitro serial transfer experiments on Q�� The
dynamics of this model�system� which contains a protein enzyme as part of the environment��
was subsequently studied in great details by Manfred Eigen and his collaborators �� �� � ��� Cat�
alytic activity of RNA for a variety of reactions has been discovered ��� �� ��� ��� including a
limited capacity for replication and self�replication ��� ���� Arti�cial molecules that self�replicate
in a protein�free environment have been synthesized� for instance� by Orgel ���� Rebek ��� ���
��� and von Kiedrowski ����� All these �ndings add to the credibility of an RNA world� as one
of the crucial stages in prebiotic evolution �����
In parallel to these advances in template chemistry a mathematical theory of molecular evolution
has been developed� based on a series of pioneering papers by M� Eigen and P� Schuster ���� ���
��� ��� Ignoring the molecular details one considers the dynamics of replicators as a auto�catalytic
chemical� reactions of the form
�A� � I � �I � �W ��
where I is the replicator� �A� denotes the energy�rich building material for replicators and �W �
indicates that the formation of the replicator will in general produce low�energy waste��
One of the central questions in any theory of molecular evolution concerns the behavior of a collec�
tion of competing �or otherwise interacting� species of replicators I�� � � � � In� While the replication
of polynucleotides is in reality an overwhelmingly complicated process involving a huge number
� � �
Happel et al�� Autocatalytic Networks with Translation
of intermediates it nevertheless has turned out that it is sensible to focus on the over�all replica�
tion reaction� In other words� one can treat the replicators as atomic� entities� and encapsulate
their chemical details into a small set of reaction rate constants ���� The system of interacting
replicators is thus completely described by the concentrations �or relative frequencies� xk of the
di�erent species Ik� In its most condensed form the logics of the RNA world is thus captured in
the autocatalytic reaction network
Ik � Il �� �Ik � Il k� l � �� � � � � n�
The dynamical system associated with the above reaction scheme is now termed replicator equation
����
�xk � xk
�� nX
j��
akjxj �Xi�j
aijxixj
�A � �R�
Originally developed as a model of prebiotic evolution� replicator equations have been encountered
since then in many di�erent �elds� populations genetics� mathematical ecology �where they occur
disguised as Lotka�Volterra equations ������ economics� or laser physics� Their properties have been
the subject of hundreds of research papers by many research groups� most prominently among them
J� Hofbauer� P� Schuster� and K� Sigmund in Vienna� The results of the �rst decade of investigation
are compiled in the book �����
Self�replication on molecular level was clearly the crucial invention at the origin of life� Almost
equally important was the invention of translation and the genetic code� In this contribution we will
not be concerned with the question why translation was invented� or how a RNA replicator that has
invented translation would conquer the RNA world ���� �� We have a much less ambitious agenda
here� We consider a reaction system that contains a number of di�erent replicator species which all
produce a gene product that carries the enzymatic activity� Our main question is the following� Is
the picture of the selection dynamics that has emerged from the studies of autocatalytic reaction
networks still valid once the catalytic activity has shifted to translation products�
� � �
Happel et al�� Autocatalytic Networks with Translation
��The Model
We consider a system of n species I�� � � � � In of replicators or genotypes and their translation
products T�� � � � � Tn� For sake of de�niteness� we may consider the genotypes as nucleic acids and
the translation products as proteins in same later stage of prebiotic evolution ���� The replication
processes involve translation products Tj of the genotype Ij as catalysts�
�A� � Ik � Tja�kj�� �Ik � Tj � �W �
for all combinations � � k� j � n� The rates of these reactions are a�kj��A�� �Ik� �Tj�� i�e� we assume
mass action kinetics� Since there are no convincing models for the kinetics of translation in a
prebiotic setting we make the most simple choice�
�B� � Ikw�k
��Tk � Ik
with rate w�k��B���Ik�� We assume that the genotypes Ik and the translation products are di�erent
types of biopolymers� thus the concentrations of the monomers ��A�� and ��B�� are independent
from each other� Consequently we will assume that the two types of polymers are subject to
di�erent degradation or removal reactions� In the mathematical model the latter are described by
out�ows �Tk� T and �IK � � respectively�
Thus the dynamical system considered here reads in its most general form
d
dt�Ik� � �Ik�
���
nXj��
a�kj��A�� �Tj��
��
d
dt�Tk� � w�
k��B�� �Ik�� �Tk� T
Note that we are still missing equations governing the concentrations of the monomers ��A�� and
��B��� and that we have not yet speci�ed the functions and T which describe the removal
processes� It is convenient to consider these as boundary conditions� under which the replication�
translation process takes place� Three types of boundary conditions have been discussed in the
literature�
��� Constant Organization assumes �i� that the monomer concentrations are bu�ered and thus
constant in time� and �ii� that the total concentration of the di�erent polymer types is kept
� � �
Happel et al�� Autocatalytic Networks with Translation
Computer
(A)(B)(solvent)
measurements
Figure �� The hypothetical Evolution Reactor is a kind of dialysis reactor with walls impermeable to polynu�
cleotides and polypeptides� The in��ow of monomers keeps the reaction mixture away from equilibrium�
It is adjusted such that the the concentrations of the monomers occur in excess in the reactor� i�e��
the monomer concentrations do not change signi�cantly over time� The out��ow of polynucleotides
and poly�peptides through two di�erent diaphragms is controlled such that total concentration of both
types of polymers is kept at constant levels� While this type of evolution reactor will not be easy to
realize in practice it establishes boundary conditions for replication�translation systems that lead to
mathematically tractable dynamical systems�
constant in time� These conditions can in principle be enforced in a sophisticated evolution
reactor as the one shown in �gure �� It will be convenient to use the e�ective rate constants
akjdef
��� ��A��a�kj and wkdef
��� ��B��w�k �
A further simpli�cation is achieved by switching to relative coordinates
xkdef
��� �Ik�X
j
�Ij� and tkdef
��� �Tk�X
j
�Tj� �
As simple calculation then shows that
�Xi�j
xiaijtj and T �Xi
wixi �
In matrix notation we obtain thus
�xk � xk ��At�k � hx�Ati� �tk � wkxk � tkhw� xi �CO�
� �
Happel et al�� Autocatalytic Networks with Translation
The state space of this model is the direct sum of the two �n � ���dimensional simplices
corresponding to the coordinates xk� k � �� � � �n� and tk� k � �� � � �n� respectively�
��� The Continuously Stirred Flow Reactor �CSTR� is the most convenient experimental setting�
A constant �ow of rate r through the system carries the monomers with concentrations a�
and b� and removes monomers and polymers at the same �ow rate r proportional to their
concentration� Using conservation of mass one immediately �nds the dynamical equations
for the concentrations of the monomers a def
��� ��A�� and b def
��� ��B��� In order to simplify the
notation we use ykdef
��� �Ik� and uk � �Tk��
�yk � yk �a�A�u�j � r�
�uk � w�kbyk � ruk
�a � �ahy�A�ui� r�a� � a�
�b � �bhw�� yi � r�b� � b�
�CSTR�
The state of the system is described by a vector �y� u� a� b� � IRn� � IR
n� � IR� � IR��
��� We assume that there is no �ux of material into or out of the system� The system is kept away
from thermodynamic equilibrium by means of an energy consuming regeneration reaction that
produces active monomers from the degradation products of the polymers� While not very
realistic for an experimental approach this type of boundary condition is very useful for the
study of pattern formation processes as it leads to meaningful reaction� di�usion models
���� �� ���� In spatially homogeneous models we obtain equations similar to the CSTR
discussed above� The rate constants of the degradation reactions are dk and dTk � respectively�
Conservation of mass ensures that ��A�� �P
k�Ik� � a� and ��B�� �P
k�Tk� � b��
�yk � yk �a�A�u�j � dk�
�uk � w�kbyk � dTk
�a � �ahy�A�ui� hd� yi
�b � �bhw�� yi � hdT � ui
�REG�
The state space is the direct sum Sn�� � Sn�� of two n�dimensional simplices�
Since all three dynamical systems describe the same chemical reaction system� although under quite
di�erent boundary conditions it is not surprising that their dynamics is quite similar� Constant
organization is by far the most tractable case� We will therefore give a complete derivation of
� � �
Happel et al�� Autocatalytic Networks with Translation
our results only for this case� Results for the CSTR and the regeneration system are often very
similar to the CO case� The details can be found in ����� It has been observed quite often that the
dynamics of replication systems in a CSTR or in a model with regeneration reaction�s� become
very similar to the constant organization case when the �ow rate r �or the reaction rates for the
regeneration steps� become small� A singular perturbation treatment of this e�ect can be found
in ���� For a partial result see ����
�� Fixed Points
Despite the fairly complicated form of our reaction scheme it is not di!cult to compute the coor�
dinates of all equilibria� Let us begin with interior equilibria� that is� with rest points at which all
species occur with non�zero concentration� We shall use the notation � for the vector with entries
��
Theorem �� Consider the reaction translation model under constant organization� Then there is
a unique interior equilibrium �"x� "t� � int �Sn � Sn� if and only if A��� is either strictly positive or
strictly negative�
Proof� Suppose A is invertible and �"x� "t� is a �xed point of �CO�� An explicit computation shows
"xk ��A����k
wk
nXi��
�
wi�A����i
and "tk ��A����kh�� A���i
�
Thus "t � intSn if and only if �A����k has the same sign for all k� Then "xk � � as well since the
coe!cients wk are all strictly positive by assumption�
Remark� If there is an isolated interior equilibrium� then A is invertible and �A����k has the
same sign for all k� Furthermore there are at most two isolated interior rest points for CSTR�
The expression for the x�coordinates of the interior rest point above strongly suggests to introduce
the matrix
B def
��� Adiag�w��
With this de�nition we may write
"x ��
h�� B���iB��� and "t �
�
h�� A���iA���
� �
Happel et al�� Autocatalytic Networks with Translation
for the location of the interior equilibrium�
On the boundary of the state space at least one coordinate is zero� It is useful to observe that
equilibria on the boundary have a particular form�
Lemma �� Suppose all translation rates are non�zero and let � � �"x� "t� be a rest point� Then
"xk � � if and only if "tk � ��
Proof� Suppose "xk � �� Then �tk � �tkhw� xi� where the scalar product hw� xi � � by assumption�
Thus "tk � �� Now suppose "tk � �� This implies � � wk"xk and thus "xk must vanish�
If some of the wk are zero� parts of be boundary consist entirely of �xed points� We will not
consider these degenerate cases any further� The non�zero coordinates of � can be obtained by
restricting the dynamical system to the variables that do not vanish in �� i�e�� to a smaller system
of the type �RT�� The theorem above can therefore be applied also to the non�zero part of a rest
point � on the boundary of the state space� All isolated rest points of �RT� are thus obtained as
the interior rest points of �RT� restricted to a subset K � f�� � � � � ng of the n replicating species�
The stability analysis of the rest points will turn out to be very complicated in general� It is
fairly easy� however� to determine the stability of a boundary equilibrium � against introduction
of species which are not present in �� The corresponding directions are called transversal ����� To
this end it will be convenient to temporarily rearrange the order of the coordinates such that
� � �x�� t�#x�� t�# � � � #xn� tn�
and "xi � "ti � � for � � i � m� The entries of the Jacobian matrix in the �rst �m rows and
columns are readily computed�
� �xk�x�
��� � �kl��A"t�k � h"x�A"ti� � �def
��� Lk����kl
� �xk�t�
��� � �
� �tk�x�
��� � �klwk
� �tk�t�
��� � �hw� "xi
Consequently the Jacobian �RT��� is of the form
� � �
Happel et al�� Autocatalytic Networks with Translation
�BBBBBBBBBBB�
�L���� �w� hw� "xi
�� � � � �
� � � � � �� �
�
�L���� �w� hw� "xi
��
� �� �
���� � �
���
� � � � �
�Lm��� �wm hw� "xi
�� �� � � � � �
X Y
�CCCCCCCCCCCA
where the ��n�m���m block X is irrelevant for the stability of � and Y is the ��n�m����n�m�
Jacobian matrix of �RT� restricted to the species that do not vanish in the equilibrium �� The
eigenvalues corresponding to the transversal direction k are now easily computed from the �� ��
blocks� We �nd explicitly
����k � Lk��� and �
���k � �hw� "xi � ��
Assuming as usual hw� "xi �� � we have "t � �hw��xidiag�w� "x and thus
����k �
�
hw� "xi��Adiag �w� "x�k � h"x�Adiag �w� "xi� �
�
hw� "xi��B"x�k � h"x�B"xi� �
Following ���� ��� we will say that a rest point � of �RT� is saturated if all eigenvalues belonging to
the transversal directions are non�positive� The above considerations can then be summarized as
Theorem �� Let � � �"x� "t� be a boundary equilibrium of a replication translation model for which
A is invertible and wk � � for all k� Then � is a saturated equilibrium if and only if "x is saturated
equilibrium of the second order replicator equation with interaction matrix B � Adiag�w�� i�e�� if
and only if
$�k���def
��� �B"x�k � h"x�B"xi � � for all k with "xk � ��
The matrix B � Adiag�w� occurs both in the explicit expression for the x�coordinates of an
interior rest point and in the expression for the transversal eigenvalues of a boundary equilibrium�
In both cases our result match the situation in a second order replicator equation with B as
interaction matrix� We will see in the following sections that this relation between �RT� and
replicator equations is even deeper�
In ���� it has been proposed to represent an autocatalytic network by a directed graph with colored
edges� In complete analogy we introduce the same notation here� The vertices of the graph %�RT �
� � �
Happel et al�� Autocatalytic Networks with Translation
b > bjjij
Commensalism Parasitism
b bjjijb b jjij= <
MutualismSymbiosis
ji i i
i i
i
j j
j
j
Amensalism
Competition
Neutralism
j
>b b
b b
b b
=
<
ji ii
ji ii
ji ii
Predator-Prey
Figure �� The di�erent types of interaction� full arrows indicate bij � bii � �� empty arrows bij � bjj ��
associated with the reaction�translation model �RT� are the replicating species I�� � � � � In� There is
a �full� arrow from j to i if bij � bii � �aij � ajj�wj � �� and there is a �empty� arrow from j to i
if bij � bjj �� As an example� look for the two�species model as follows� The six di�erent graphs
of this type on n � � vertices match the usual classi�cation of ecological interactions between two
species� see �gure �� In the following section we show that they also correspond in a very natural
way to the classi�cation of the dynamical behavior of the two�species replication�translation system�
see �gure ��
��Two Species
The special case n � � allows for a complete analysis of the dynamics under constant organization�
We consider here the general two�species system with general selection matrix A with entries
aij � � and and translation constants �w�� w�� � �� It will be convenient to use the following
abbreviations�
c�def
��� a�� � a�� � c�def
��� a�� � a�� �
The results of the previous section imply immediately that an interior equilibrium exists if and
only if c�� c� � � or c�� c� �� The Jacobian of the interior rest point can be readily diagonalized
� � �
Happel et al�� Autocatalytic Networks with Translation
with the help of Mathematica� One �nds the two external eigenvalues
�ext� �D
��c� � c��and �ext� � �
w�w��c� � c��
c�w� � c�w�
which do not in�uence the dynamical behavior on S� � S�� Both are negative whenever there is
an interior rest point� The dynamical stability of this �xed point is determined by the remaining
two eigenvalues
�� ���c� � c��w�w�
p�c� � c���w�
�w�� � �c�c�w�w��c�w� � c�w��
��c�w� � c�w���
R
RT :
:
(x ,t ) (x ,t )
(x )
(x ,t )(x ,t )
(x )1 2
1 1 1
2 1 2 2
2
Figure �� Comparison of the phase portraits of �RT� and �R�� Black circles indicate sinks� gray ones saddle�points�
and white ones are sources� The corresponding graph %�RT � are shown below�
This is of the form �p�� � �� where � is always negative and sgn� � sgnc�� provided c� and c�
have the same sign� Assume that c�� c� � �� Then the square root is either complex or if it is real
then it is smaller than �� Consequently �� � and the interior rest point is a sink� If c�� c� � �
then � � and the square root is real and larger than �� hence � � and � � �� and the interior
equilibrium is a saddle point�
Finally� let us brie�y consider the ��species equilibria� Their stability is determined by the eigen�
values��� �# �� �� � �� � �
a��c�c� � c�
� �� � �c�c�
c� � c��
��� �# �� �� � �� � �a��c�c� � c�
� �� � �c�c�
c� � c��
Summarizing our calculations we have
� �� �
Happel et al�� Autocatalytic Networks with Translation
c�� c� � Both corners are sinks and the interior equilibrium is a saddle point� Its unstable
manifold connects to both sinks�
c� � �� c� � Corner � is a sink and corner � a saddle point� The unstable manifold of corner �
connects to the sink� There is no interior rest point�
c� �� c� � � Corner � is a sink and corner � a saddle point� The unstable manifold of corner �
connects to the sink� There is no interior rest point�
c�� c� � � Both corners are saddle points and the interior equilibrium is a sink� The unstable
manifolds of both corner saddle points connect to the interior sink
The phase portraits corresponding to these four cases are shown in Fig� �� They compare directly
to the phase portraits of the second order replicator equation with interaction matrix A�
��Barycentric Transformation
Before we proceed with the linear stability analysis of interior equilibria in larger systems� we
brie�y discuss a transformation that will turn out to be a crucial tool for most of our results� The
following lemma is well known� see e�g� ���� sect� �����
Lemma �� Let D � diag�d�� � � � � dn� be a diagonal matrix with di � �� Then there is a di�eo�
morphism B � Sn � Sn mapping the phase portrait of the second order replicator equation with
interaction matrix A to the phase portrait of the second order replicator equation with interaction
matrix AD�
This result can used to simplify the algebra for the stability analysis of an interior rest point�
Suppose there is an interior rest point "x� Setting didef
��� � "xi sends "x results in B�"x� ��n�� i�e��
the interior rest point is mapped to the barycenter of the simplex� Therefore B is usually called
barycentric transformation� The fact that the coordinates of the interior rest point are now of a
very simple form simpli�es the algebraic manipulations in many cases� Fortunately� a similar result
holds for our replication�translations system �RT��
� �� �
Happel et al�� Autocatalytic Networks with Translation
Theorem �� Let aij be an arbitrary matrix� �x� t� � Sn � Sn and let �c� d� � int �Sn � Sn�� Then
B � Sn � Sn � Sn � Sn� �x� t� � �u� v� de�ned by
ui �cixiPj cjxj
and vi �ditiPj djtj
is a di�eomorphism mapping the orbits of
�xk � xk
��X
i
akiti �Xi�j
xiaijtj
�A
�tk � wkxk � tkXi
wixi
onto the orbits of
�uk � uk
��X
i
bkivi �Xi
uiXj
bijvj
�A
�vk �
���kuk � vk
Xj
�juj
�A ��u� v��
where the coecients of the latter di�erential equations are bij � aij dj� �i � diwi ci� and
��u� v� �
Xl
vldl
��Xl
ulcl
� ��
Proof� The inverse map B�� is given by
xi �uici
�Pj c
��j uj
and ti �vidi
�Pj d
��j vj
�
Di�erentiating ui and vi yields
�ui ���P
j cjxj
�� ��� �xiciX
j
cjxj � cixiXj
cj �xj
�� �
� ui
��X
j
aijtj �Xj
xjXl
ajltl
�A �
� ui
��X
j
aijdj
vj �Xj
ujXl
ajldlvl
�A �P
j d��j vj
�vi �di �tiPj djtj
� viXj
dj �tjPl dltl
�diwixiPj djtj
� viXj
djwjxjPl dltl
�
��diwi
ciui � vi
Xj
djwj
cjuj
�APj d
��j vjP
j c��j uj
By means of a change of velocity and setting bij � aij dj and �i � diwi ci we obtain �nally the
proposition�
� �� �
Happel et al�� Autocatalytic Networks with Translation
Corollary �� Let "� � �"x� "t� be an equilibrium of �CO� and let
ui �xi
"xiP
j "x��j xj
and vi �ti
"tiP
j"t��j tj
�
Then the �xed point of the dynamical system obtained from �CO� by a barycentric transformation
with parameters ui� vi is "B�"�� ��n������
Proof� From theorem � one �nds immediately
"ui �"xi
"xiP
j "x��j "xj
��
nand "vi �
"ti"tiP
j"t��j"tj��
n�
��Competitive Systems
Schl�ogl ���� investigated two model systems in which the substance Xi is formed from a substrate A
via �rst order and second order autocatalysis� respectively� A�Xi � �Xi ��rst order autocatalysis�
and A � �Xi � �Xi �second order autocatalysis�� In the �fully� competitive case the translation
products catalyze only the replication of their own gene� i�e�� the interaction matrix is diagonal
A � diag�k�� k�� � � � � kn�� Therefore the di�erential equations for the competitive model simplify to
�xk � xk
��kktk �X
j
kjxjtj
�� �tk � wkxk � tk
Xj
wjxj
The phase portraits of this dynamical system as equivalent with the phase portraits of
�uk � ui
��vi �X
j
ujvj
�� �vk � �uk � vk���u� v�
as a consequence of the barycentric transformation�
The Jacobian matrix of this vector �eld is readily computed�
�
�uj�ui � �uivj
�
�vj�ui � �uiuj � ui�ij
�
�uj�vi � �ui � vi�
��
�uj� �ij��u� v�
�
�vj�vi � �ui � vi�
��
�vj� �ij��u� v�
� �� �
Happel et al�� Autocatalytic Networks with Translation
At the �xed point P � �n����� we �nd that ��P� � w� Since ui � vi at this point� the values of
�� �ui and �� �vi are irrelevant� The Jacobian is of the form
�f�P� �
�A BC D
�
where each of the four quadratic matrices A� B� C� and D is circulant� We �nd explicitly�
A � ��
n�J � B � wI � C � �
�
n�J �
�
nI � and D � �wI �
where J is the matrix with all entries one and I is the identity matrix� Matrices of this type can
be analyzed using the following interesting result�
Theorem �� Let M be a mn �mn matrix which has m� circulant blocks Mij of size n� n� The
vectors ��k� with entries
��k�j � exp����
j
k� � for k � �� � � � � n� � and j � � � � �n
are well known to be eigenvectors of any circulant n � n matrix� Set Mij��k� � �
�k�ij ��k�� let R�k�
be the m �m matrix with entries ��k�ij � and denote the eigenvalues of R�k� by &�k�� �
Then &�k�� is an eigenvalue of M � In particular� if all matrices R�k� are diagonalizable� then we
obtain all eigenvalues of M as eigenvalues of the matrices R�k��
Proof� We rely on the fact that the ��k� form a basis of eigenvectors for all circulant matrices�
Let z � IRm and suppose � � z � ��k� is an eigenvector of M � Then
M� �
�BB�
M��z���k� M��z��
�k� � � � M�mzm��k�
M��z���k� M��z��
�k� � � � M�mzm��k�
������
���Mm�z��
�k� Mm�z���k� � � � Mmmzm�
�k�
�CCA � �R�k�z�� ��k� � &�k���
Since ��k� is non�zero� this equation is equivalent to the eigenvalue equation for the matrix R�k��
Thus � is an eigenvector of M if z is an eigenvector of R�k�� and all eigenvalues of R�k� are also
eigenvalues of M �
Now suppose that all matrices R�k�� � � k � m are diagonalizable� Denote by z�k���� � � � n� an
orthonormal eigensystem of R�k� and let ��k��� def
��� z�k��� � ��k� be the corresponding eigenvectors
of M � We �nd that
hz�k��� � ��k�� z�k����� � ��k��i � hz�k���z�k
�����i h��k���k��i � ����� � �k k� �
� � �
Happel et al�� Autocatalytic Networks with Translation
i�e�� all eigenvectors corresponding to di�erent k are orthogonal� even if the corresponding eigenval�
ues should coincide by chance� Thus the ��k��� form a complete orthonormal basis of eigenvectors
for M �
In the terminology of theorem we �nd
R��� �
�� �
n �w �w
�and R�j� �
�� �
nw �w
�
for the competitive model� R��� belongs to the external directions since ���� � �� The correspond�
ing eigenvalues are
&����� � ��
nand &����� � �w�
The remaining eigenvalues &�j��� for � � j n and � � �� � are now easily computed using
theorem �
&��� � �w
��
�
pw� � w n
independent of j� Both &��� and &��� are n � ��fold degenerate� and it is easy to check that
&��� � &��� for all n � � and w � �� Thus � is a saddle point with n�� unstable eigenvalues�
The result immediately carries over to all equilibria on the boundary of the state space in the
following form� Let � be a rest point on the boundary and suppose there are m non�vanishing
species at �� Then � has m � � positive and m � � negative eigenvalues which all belong to the
directions spanned by the non�vanishing species� In particular� if � is stable� then m � �� i�e��
only the single�species equilibria can be stable� That they are in fact stable is determined by the
transversal eigenvalues �k��� � �w ��
��Cooperation
The hypercycle ���� may serve as the paradigm of a cooperative system� Each translation product
catalyzes the replication of one other species in a circular arrangement� see �gure # consequently�
the interaction matrix is circulant� and the corresponding system of di�erential equations reads�
�xk � xk
��kktk�� �
Xj
kjxjtj��
��
�tk � wkxk � tkXj
wjxj �
� �� �
Happel et al�� Autocatalytic Networks with Translation
Again a barycentric transformation yields
�uk � ui
��vi �X
j
ujvj��
��
�vk � �uk � vk���u� v� �
The external eigenvalues of the interior equilibrium ����� are
&����� � ��
nand &����� � �w �
For the remaining eigenvalues we �nd
&���j � �
w
��
�
rw� �
w
ne���j��n �
and their real parts can be readily computed�
�&j� � �w �
rw
�
sw �
ncos��
rw� �
�w
ncos� �
�
n��
It is not complicated now to derive the critical value of w at which a Hopf�Bifurcations occurs�
��
w
�crit
� ��
n
sin�����n� ��j
n
�
cos
����n� ��j
n
� �
Tn 1
I2
T1I
T2
I
T3
3
I4
In
a)
1
2
3
4
n
b)
Figure �� The hypercycle� a� chemical reaction scheme� b� graph representation�
� � �
Happel et al�� Autocatalytic Networks with Translation
Since w � �� only values of j in the interval n j �n can ful�l this condition� Consequently�
there are no Hopf bifurcations in the two�species model� For larger systems we �nd
n � �� j � � wcrit � � ��
n � � j � � wcrit � �� Since we consider only non�zero values of the translation rate� there is in
fact no stable �xed point� The corresponding hypercyle equation� however� exhibits
a marginally stable �xed point �����
n � � There are no stable interior �xed points� Computational studies indicate that there is
a �globally� stable limit cycle� as in the case of the second order replicator equations
�����
All equilibria on the boundary are non�saturated and degenerate� in complete analogy to the
situation in the elementary hypercycle ����
�Reduction to Replicator Equations
A good introduction to singular perturbation theory are the books ���� ��� ���� The results outlined
in the following few paragraphs are well known� see e�g�� ���� �� ��� ���� We compile them here in
order to make this contribution self�consistent� Consider the singularly perturbed problem
�x � f�x� y� a� ��
� �y � g�x� y� a� ���SPP �
where x � X and y � Y � a � A � IRp is the admissible space of parameters� and � � I � R�
Furthermore let K � X � Y be compact such that int K is simply connected� We are interested
in the dynamics in the compact set K�
Suppose g has the following properties�
�i� There is a unique function � � X �A� Y such that g�x� ��x� a�� a� �� � ��
�ii� The Jacobian J�x� a� ��g
�y�x� ��x� a�� a� �� is uniformly hyperbolic on A � X� i�e�� there is
a positive constant � � � such that absolute value of all eigenvalues of J�x� a� is bounded
below by � for all x � X and all a � A�
�iii� For �xed a � A we have f�x� ��x� a��jx � Xg �K �� ��
� �� �
Happel et al�� Autocatalytic Networks with Translation
Theorem �� Under the above hypotheses there are open sets A� � A� I� � I such that for all
�a� �� � A��I � there is a unique integral manifoldMa�� � fy � ��x� a� ��jx � Xg with the following
properties
�i� � � X � A� � I� � Y is continuously di�erentiable�
�ii� � satis�es uniformly on X � A�
lim���
��x� a� �� � ��x� a� and lim���
��
�x�x� a� �� �
��
�x�x� a� �
If J�x� a� is stable on X�A then the long�time behavior of a trajectory passing through a point x�
in a suitably small neighborhood ot the integral manifoldMa�� is determined by the dynamics on
this manifold� i�e�� by the di�erential equation �x � f�x� ��x� a� ��� a� ��� Under these circumstances
we introduce the notation
F �x� a� def
��� f�x� ��x� a�� a� �� �
'�x� a� �� def
��� f�x� ��x� a� ��� a� ��� f�x� ��x� a�� a� �� �
The di�erential equation �x � F �x� a� is known as the degenerate system� Property �ii� of � means
that there is a continuous function ���� with ���� � � such that k��x� a� �� � ��x� a�k� �����
Here k � k� denotes the C� norm� see� e�g� ���� p� ���� If f is continuously di�erentiable with
uniformly bounded derivatives on X � Y there is a continuous function ����� with ����� � � such
that k'�x� a� ��k� ������ In other words� the dynamics of trajectories near the integral manifold
Ma�� is described by the di�erential equation
�x � F �x� a� � '�x� a� ��
where '�x� a� �� is a regular perturbation of the degenerate system �x � F �x� a�� In such a case we
will say the the singularly perturbed problem �SPP� reduces to the degenerate problem�
These facts from singular perturbation theory make precise in what sense our replicaton�translation
system is related to replicator equations� We set wk � �k � and consider the limit wk � �� i�e�
�� � with constant �k and �nd the following very general correspondence�
Theorem � The replication translation model �CO� reduces in the singular limit wi �� to the
second order replicator equation with interaction matrix bijdef
��� aij�j�
� �� �
Happel et al�� Autocatalytic Networks with Translation
0.1 0.2 0.3 0.40.0
0.2
0.4
0.6
0.80.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
0.1 0.2 0.3 0.40.00
0.20
0.40
0.60
0.800.1 0.2 0.3 0.4
0.00
0.20
0.40
0.60
0.80
a) b)
c) d)
Figure �� The close relations between �R� and �RT� are perhaps best exempli�ed by strange attractors�
Chaotic attractors have been found in three�species Lotka�Volterra equations ��� ��� �� �� ���� Using
Hofbauer�s transformation ���� they can be translated to four�species second order replicator equations�
A two�parameter family of strange attractors with interaction matrix
A��� �� �
�B�
� ���� ����� ���� � ���� ��� � ���������� ����� � ��� � ���� ������������ ������ �� ���� � �������� � �� ���� ��� �� ����� ���� � ����� �
�CA
has been studied in detail � ��� The numerical examples shown here correspond to � � ����� � � �
and wk � w for all k�
The graphs show a two�dimensional projection of the x�coordinates� a� slow translation w � ���� b�
w � �� c� very fast translationw � ��� and d� second order replicatior equation�
Proof� Let wk � �k � with �k� � � �� With the notation we obtain the singular perturbation
problem
�xk � xk
��X
i
akiti �Xi
xiXj
aijtj
�A
� �tk � �kxk � tkXi
�ixi
The fast time scale yields tk�x� ��kxkPi �ixi
in the limit � � �� It is easy to see that this solution
is stable for all x�
� �tk�tj
��t�x�
� ��
��kjXi
�ixi�
� �� �
Happel et al�� Autocatalytic Networks with Translation
and hence the singular perturbation problem reduces to
�xk ��P
j �jxjxk
��X
i
aki�ixi �Xi�j
aij�jxjxi
�� �
Using bij � aijwj and a change in velocity yields the second order replicator equation with inter�
action matrix B�
Remark� A barycentric transformation shows that the replicator equation �R� has qualitatively
the same phase portraits since the constants �j are all positive�
Little can be said in general about the behavior for small translation rates� The extremal models
discussed in sections and � show that bifurcations leading away from replicator like� behavior
can occur for slow replication�
�Replication� Translation� and Mutation
Mutation is an integral component of any evolving system� Since the e�ect of mutation is to
introduce new� hitherto unknown species into the system it cannot be adequately modeled within
the framework of di�erential equations if the number of possible species is very large� Nevertheless�
it makes sense to consider the e�ect of mutations between the species in our reaction system�
Mutation can be viewed as a perturbation that pushes the �ow of the replicator equation away
from the boundary into the interior of the state space ����� Assuming both incorrect replication
and incorrect translation we have to deal with the following reaction network
�A� �Xk � TlQkmakl�� Ik � Im � Tl
�B� � IkQTkmwk�� Ik � Tm
The accuracy of the replication step is described by the probabilities Qmk of forming a mutant Im
from a template Ik� Correspondingly� a translation product of type Tm is produced from a gene
Ik with probability QTmk� The kinetic equations read
�xk �nX
j��
Qkjxj
nXi��
ajiti � xk
�tk �nX
j��
QTkjwjxj � tk
T
� �� �
Happel et al�� Autocatalytic Networks with Translation
where the �uxes and T are adjusted such thatP
j xj � � andP
j tj � �� The terms in the
above equation can be rearranged in order to emphasize the perturbational nature of mutation in
this model��xk � xk ��At�k � hx�Ati� �
Xj ��k
�Qkjxj�At�j �Qjkxk�At�k�
�tk � wkxk � tkhw� xi�Xj ��k
�QTkjwjxj � QT
jkwkxk� �RTM �
For sake of de�niteness we assume that the mutation matrix Q is of the form
Q �
�BB��� �n � ��� � � � � �
� �� �n� ��� � � � ����
������
� � � � � �� �n � ���
�CCA
An analogous representation is assumed for QT � with a translation error rate � replacing the
mutation rate �� Then �RTM� simpli�es to
�xk � xk ��At�k � hx�Ati� � �P
j ��k �xj�At�j � xk�At�k��tk � wkxk � tkhw� xi � �
Pj ��k �wjxj �wkxk�
Let us use the notation � � �x� t� � Sn � Sndef
��� S� Then the replication�translation model with
mutation takes the form
�� � RT ��� �M��# �� ��
where RT is the error�free replication�translation part andM describes the e�ects of mutation� For
the full model �RTM� even the computation of the equilibrium points can be done only numerically
except for a few special cases discussed below� The perturbation methods described in ���� are
applicable here�
First we need some notation for the subsimplices that constitute the boundary of the state space
S� Let (FK � Sn be the subsimplices on which xk � � for all k � K� where K is an index set which
is neither f�� � � � � ng nor empty� The relative interior of the sub�simplex (FK will be denoted by
FK �
Denition �� A vector �eld M��# �� �� on S is a mutation �eld if it has the following properties
�i� M � �Mx�Mt� is continuously di�erentiable and kMk� is bounded on S � I � J � where I
and J are non�empty intervals I� J � IR�� �
�ii� M��# �� �� � � for all � � S�
�iii� Mxk��# �� �� � � if xk � � and M
tk��# �� �� � � if tk � � for all ��� �� � �I � J� n f��� ��g�
We say that M is a strong mutation �eld if the weak inequalities in condition �iii� are replaced by
strong inequalities�
� �� �
Happel et al�� Autocatalytic Networks with Translation
Lemma �� �i� (FK � (FK is invariant under the ow of �RT�� �ii� (FK � Sn is invariant under the
ow of �RTM� with � � � and arbitrary ��
Proof� �i� Let k � K� From xk � tk � � we infer immediately that �xk � �tk � � as well� thus
(Ffkg � (Ffkg is invariant� Repeating this argument for all k � K implies the proposition�
�ii� If � � � and � � � we �nd �tk � � if tk � � and xk � � �� �xk � �� Thus (FK �Sn is invariant�
A direct analogue of the rest�point migration theorem which is proven in ���� for selection�mutation
equations can be obtained for �RTM�� and even more general versions of models of erroneous
replication and translation� The proof of the following result parallels the selection�mutation
version line by line� therefore we omit the details here�
Theorem �� Let "� � (FK � (FK be a rest point of �RT�� let M��# �� �� be a strong mutation �eld�
Then there is an �� � � such that for all � � � �� and all � � � � �� the following statements
are true
�i� If the rest point "� is regular then ���� � intS if and only if the transversal eigenspace ET �"��
is stable� i�e�� if "� is saturated�
�ii� If the transversal eigenspace ET �"�� has at least one positive eigenvalue� say �l � �� then all
�xed points �j��� derived from "� lie outside the state space S�
a) b) c)
d) e) f)
Figure � The development of the �xed points of the competitive model for various values of �� where � � � for
n � �� a� no mutation� b� subcritical values for �x� c��f� supercritical value for �x� � �Sink� � � Saddle
� �� �
Happel et al�� Autocatalytic Networks with Translation
The consequences of the rest point migration theorems are dramatic� Mutational perturbations
drive all non�saturated rest points into the non�physical exterior of the state space� The dynamics
of the systems are thus simpli�ed due to the reduction in the number of the �xed points� This is
the same behavior as has been observed for the selection mutation models in ����� Figure shows
an exceptional case� all �xed points are saturated in this model and are therefore pushed into the
interior of the state space� The hypercyclic model discussed in section �� on the other hand� is left
with a unique rest point for small � � ��
Non�perturbative results can be obtained only for very simple interaction matrices� This matches
the situation for selection�mutation equations discussed in ����� A few results can be obtained for
circulant interaction matrices A and equal translation rates wk � w for all k�
Lemma �� Suppose A �� � is circulant and all translation rates are equal to w � �� Then
"� � �"x� "t� � �n����� is the unique interior equilibrium of �RTM��
Proof� Since A is circulant� �A� "t�j � � �j� Thus h"x�A"ti � �h"x��i � � and hw� "xi � w�
Substituting this into �RTM� immediately yields the lemma�
Let us now consider two special cases in more detail� The symmetric competitive model has
interaction matrix A � kI� The di�erential equation �RTM� reduces to
�xk � k � xk �tk � hx� ti� � � k
����n � ��xktk �X
j ��k
xjtj
��
�tk � w�xk � tk� � � ���n � �� � wxk�
The eigenvalues at the interior equilibrium �n ����� are readily computed� For the two external
eigenvalues we �nd &��x � �k n and &��t � �w� For the remaining eigenvalues one �nds
&j� ��k�� w
��
�
r�n � k��� �
kw
n��� n����� n��
The critical values of � and � for which the interior �xed point becomes stable are readily obtained
from the above expression�
�crit �n� � �
n�n� � ��and �crit �
�n�� �
n�n�� ���
The stability of the unique interior rest point �n ����� can be determined also for the symmetric
cooperation model with interaction matrix akj � �i�j��� The Jacobian has the same external
� �� �
Happel et al�� Autocatalytic Networks with Translation
eigenvalues as in the case of the symmetric competition model discussed above� For the physically
relevant eigenvalue we �nd
&j� � �k� �w
��
�
r�w � k��� �
kw
n�� � n����� n�� exp�����
j
n�
The critical values of the mutation rate can be computed from the above equation as solutions of
a cubic equation� In the case n � � the analytical solution is tractable but too messy to be printed
here� With the help of Mathematica we could show that if �n����� is a stable equilibrium for � � ��
then it remains stable for arbitrary values of �� For higher dimensions numerical studies show that
the interior rest point becomes stable for large enough mutation rates�
The behavior of �RTM� is in general very similar to the behavior of a second order replicator
network under the in�uence of mutation� Increasing mutation rates force the �ow towards the
middle� of the state space� and very high mutation rates lead yield a globally stable interior
equilibrium ���� ����
���Discussion
In this contribution we have compared the dynamics of the most simple replication�translation
model �RT� and the dynamics of second order replicator equations �R�� The version of �RT�
considered here is the simplest extension of the replicator equation model that separates the repli�
cating entities �genomes� from the catalysts mediating the replication process� We �nd that the
behavior of such a system is very closely related to the dynamics of an autocatalyic replication
process which does not involve translation� More precisely�
� The replication�translation model with rate constants aij for the replication of a genome of
type i using the catalytic activity of translation product j and with a translation rate wj
for the translation of a genome of type j corresponds to a second order replicator equation
�R� with interaction coe!cients bij � aijwj � We have shown that the replicator equation
with these interaction coe!cients is in fact the singular perturbation limit of �RT� for fast
translation�
� The dynamics of �R� and �RT� are also closely related for small translation rates� In
particular� the there is an interior rest point of �RT� if and only if the corresponding replicator
� � �
Happel et al�� Autocatalytic Networks with Translation
equation has an interior equilibrium� Furthermore� a rest point of �RT� is saturated if and
only if the corresponding equilibrium of �R� is saturated�
� The dynamics of a two�species system is the same in both models�
� Mutations a�ect both models in qualitatively the same way�
Not surprisingly� non�generic features of second�order replicator equations are not shared by the
replication�translation model� For instance� we �nd super�critical Hopf�bifurcation in three�species
models which do not occur in second order replicator equations� It is interesting to note in this
context that a monotonic distortion of the linear�response functions �Ax�k in a the second order
replicator equation can have the same e�ect ����
Of course� our model of translation is unrealistically simple� it does not consider the formation
of a complicated translational apparatus such as a ribosome� nor does it contain any regulatory
mechanism� But then� we have never claimed to model a realistic early translation mechanism��
The purpose of this contribution is to show that networks that are based on replication can be
�roughly� described by replicator equations even if the replicating entities� in our case the genotypes
together with their translation products� have non�trivial internal structure� A similar question
was raised in ��� �� for quite di�erent variations of the replicator theme�
This investigation is thus only a �rst step towards a much more ambitious goal� Given a chemical
reaction network can we devise a coarse grained description in which the players are collections of
chemical species� such that the dynamics of the original reaction network is well simulated by a
dynamics on the coarse grained level�
References
��� A� Arneodo� P� Coullet� J� Peyraud� and C� Tresser� Strange attractors in volterra equations
for species competition� J� math �Biol�� ���������� �����
��� A� Arneodo� P� Coullet� and C� Tresser� Occurance of strange attractors in three� dimensional
volterra equations� Phys� Lett�� ��A�������� �����
��� G� J� Bauer� Traveling waves of in vitro evolving RNA� Proc� of the nat� academy of sciences�
����������� �����
� �� �
Happel et al�� Autocatalytic Networks with Translation
�� C� K� Biebricher� M� Eigen� W� C� Gardiner jr�� and R� Luce� Kinetics of RNA replication�
Biochemistry� ����������� �����
��� C� K� Biebricher� M� Eigen� W� C� Gardiner jr�� and R� Luce� Kinetics of RNA replication�
Competition and selection among self�replicating RNA species� Biochemistry� ���������
����
�� C� K� Biebricher� M� Eigen� W� C� Gardiner jr�� and R� Luce� Kinetics of RNA replication�
Plus�minus asymmetry and double strand formation� Biochemistry� ���������� �����
��� T� Cech� RNA as an enzyme� Sci�Am�� ������� November ����
��� T� R� Cech� Conserved sequences and structures of group I introns � building an active site
for RNA catalysis � a review� Gene� ����������� �����
��� J� Doudna� S� Couture� and J� Szostak� A multi�subunit ribozyme that is a catalyst of and
and a template for complementary�strand RNA synthesis� Science� ������������ �����
���� J� Doudna� N� Usman� and J� Szostak� Ribozyme�catalyzed primer extension by trinucleotides�
A model for the RNA�catalyzed replication of RNA� Biochemistry� ������������� �����
���� M� Eigen� Selforganization of matter and the evolution of macromolecules� Naturwiss�� �����
���� �����
���� M� Eigen and P� Schuster� The hypercycle� A� emergence of the hypercycle� Naturwiss��
������� �����
���� M� Eigen and P� Schuster� The hypercycle� B� the abstract hypercycle� Naturwiss�� ������
�����
��� M� Eigen and P� Schuster� The hypercycle� C� the realistic hypercycle� Naturwiss�� ��������
�����
���� M� Eigen and P� Schuster� The Hypercycle� Springer�Verlag� New York� Berlin� �����
��� M� Eigen� P� Schuster� K� Sigmund� and R� Wol�� Elementary step dynamics of catalytic
hypercycles� BioSystems� �������� �����
���� M� Famulok� J� Nowick� and J� Rebek Jr� Self�Replicating Systems� Act�Chim�Scand��
������� �����
� � �
Happel et al�� Autocatalytic Networks with Translation
���� N� Fenichel� Geometric singular perturbation theory for ordinary di�erential equations� J�
Di�� Eqns�� ��������� �����
���� R� Gesteland and J� Atkins� editors� The RNA World� Cold Spring Harbor Laboratory Press�
Cold Spring Harbour� NY� USA� �����
���� M� E� Gilpin� Spiral chaos in a predator pray system� Amer� Nat�� ����������� �����
���� K� H� Gordon� Were RNA replication and translation directly coupled in the RNA ��protein��
world� J� Theor� Biol�� ������������ �����
���� C� Guerrier�Takada and S� Altman� Catalytic activity of an RNA molecule prepared by
transcription in vitro� Science� ����������� ����
���� R� Happel� Dynamics of Autocatalytic Reaction Networks� Replication with Translation�
Master�s thesis� University of Vienna� ����
��� R� Happel and P� Stadler� Autocatalytic Replication in a CSTR and Constant Organization�
subm� to J� math� biol�� �����
���� R� Hecht� Replicator networks with intermediates� PhD thesis� University of Vienna� ����
��� D� Henry� Geometric Theory of semilinear parabolic equations� volume �� of Lecture Notes
in Mathematics� Springer�Verlag� Berlin� �����
���� M� W� Hirsch and S� Smale� Di�erential Equations� Dynamical Systems� and Linear Algebra�
Academic Press� Orlando FL� USA� ����
���� J� Hofbauer� On the occurrence of limit cycles in Volterra�Lotka equations� Nonlin� Anal��
������������ �����
���� J� Hofbauer� Saturated equilibria� permanence� and stability for ecological systems� In Pro�
ceedingd on the Second Autumn Course on Mathematical Ecology� Trieste� Italy� ����
���� J� Hofbauer� J� Mallet�Paret� and H� L� Smith� Stable periodic solutions for the hypercycle
system� J� dyn� and di�� equ�� ������� �����
���� J� Hofbauer and K� Sigmund� Dynamical Systems and the Theory of Evolution� Cambridge
University Press� Cambridge U�K�� �����
� �� �
Happel et al�� Autocatalytic Networks with Translation
���� H� Knobloch and B� Aulbach� Singular perturbations and integral manifolds� J�Math�Phys�Sci�
�������� ����
���� J� Murray� Asymptotic Analysis� Springer� New York� Berlin� Heidelberg� Tokyo� ����
��� L� Orgel� Molecular Replication� Nature� �������� �����
���� J� R�E� O�Malley� Singular perturbation Methods for Ordinary Di�erntial Equations� Springer�
Verlag� �����
��� J� Rebek Jr� Synthetic Self�replicating Molecules� Sci�Am�� ��������� ����
���� F� Schl�ogl� Chemical Reaction Models for Non�Equilibrium Phase Transitions� Z� Physik�
������ � ��� �����
���� W� Schnabl� P� F� Stadler� C� Forst� and P� Schuster� Full characterization of a strange
attractor� Physica D� ������� �����
���� K� R� Schneider� Singularly perturbed autonomous di�erential systems� In H� Bothe� W� Ebel�
ing� A� Kurzhanski� and M� Peschel� editors� Dynamical Systems and Environmental Models�
VEB�Verlag� Berlin� �����
��� P� Schuster� Mechanisms of molecular evolution� In S� Fox� editor� Selforganization� pages
������ Adenine Press� ����
��� P� Schuster and K� Sigmund� Replicator dynamics� J�Theor�Biol�� ������������ �����
��� P� Schuster and K� Sigmund� Dynamics of evolutionary optimization� Ber�Bunsen�
Gesellsch�phys�Chem�� �������� �����
��� P� Schuster� K� Sigmund� and R� Wol�� Dynamical systems under constant organisation
I�topologigal analysis of a family of non�linear di�erential equations � a model for catalytic
hypercycles� Bull� Math� Biol�� �������� �����
�� J� M� Smith and E� Szathmary� The Major Transitions in Evolution� W�H�Freeman� Oxford�
New York�Heidelberg� �����
��� J� S�Novick� Q� Feng� and T� Tjivikua� Kinetic studies and modeling of a self�replicating
system� J� Am� Chem� Soc�� ������������� �����
� �� �
Happel et al�� Autocatalytic Networks with Translation
�� S� Spiegelman� An approach to the experimental analysis of precellular evolution� Quaterly
Review of Biophysics� ��������� �����
��� B� M� R� Stadler and P� F� Stadler� Dynamics of small autocatalytic reaction networks III�
Monotonous growth functions� Bull�Math�Biol�� �������� �����
��� P� F� Stadler� Complementary replication� Math�Biosc�� ����������� �����
��� P� F� Stadler� W� Fontana� and J� H� Miller� Random catalytic reaction networks� Physica D�
���������� �����
���� P� F� Stadler and J� C� Nu$no� The in�uence of mutation on autocatalytic reaction networks�
Math�Biosci�� ����������� ����
���� P� F� Stadler� W� Schnabl� C� V� Forst� and P� Schuster� Dynamics of autocatalytic reaction
networks II� Analytically treatable special cases� Bull�Math�Biol�� �������� �����
���� P� F� Stadler and P� Schuster� Dynamics of autocatalytic reaction networks i� Bifurcations�
permanence and exlusion� Bull�Math�Biol�� ���������� �����
���� P� F� Stadler and P� Schuster� Mutation in autocatalytic networks � an analysis based on
perturbation theory� J� Math� Biol�� ���������� �����
��� C� Streissler� Autocatalytic Networks Under Di�usion� PhD thesis� University of Vienna� �����
���� A� Tichonov� Systems of di�erntial equations with a small parameter at the derivatives� Mat�
Sbornik� ���������� �����
��� C� Tresser� Homoclinic orbits for �ows in R� J� Phys� France� ������ ����
���� R� R� Vance� Predation and resource partitioning in one predator � two pray model commu�
nities� Amer� Nat�� ������������ �����
���� G� von Kiedrowski� Minimal replicator theory I� Parabolic versus exponential growth� In
Bioorganic Chemistry Frontiers� Volume �� pages ������ Berlin� Heidelberg� ����� Springer�
Verlag�
���� E� Weinberger� Spatial stability analysis of eigen�s quasispecies model and the less than �ve
membered hypercycle under global population regulation� Bull� math� biol�� �����������
�����
��� F� H� Westheimer� Polyribonucleic acids as enzymes�news and views� Nature� ������� ����
� �� �