47
Autocatalytic Networks with Intermediates I: Irreversible Reactions Robert Hecht Robert Happel Peter Schuster Peter F. Stadler SFI WORKING PAPER: 1996-05-024 SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views 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 external faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or funded by an SFI grant. ©NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure timely distribution of the scholarly and technical work on a non-commercial basis. Copyright and all rights therein are maintained by the author(s). It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only with the explicit permission of the copyright holder. www.santafe.edu SANTA FE INSTITUTE

Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

  • Upload
    lynga

  • View
    246

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Autocatalytic Networks withIntermediates I: IrreversibleReactionsRobert HechtRobert HappelPeter SchusterPeter F. Stadler

SFI WORKING PAPER: 1996-05-024

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

Page 2: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Autocatalytic Networks with Intermediates I�

Irreversible Reactions

Robert Hechta�d� Robert Happela

Peter Schustera�b�c and Peter F� Stadlera�b��

aTheoretische Biochemie� Institut f�ur Theoretische Chemie

Universit�at Wien� Vienna� Austria

bSanta Fe Institute� Santa Fe� NM

cInstitut f�ur Molekulare Biotechnologie� Jena� GermanydCurrent Address� Siemens �Osterreich� PSE�TN���� Vienna� Austria

�Mailing Address�Peter F Stadler� Inst� f� Theoretische Chemie� Univ� Wien

W�ahringerstr� ��� A�� Wien� AustriaPhone� �� � � ��� ��� Fax� �� � � ��� ��

E�Mail� studla�tbi�univie�ac�at

Page 3: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

Abstract

A class of autocatalytic reaction networks based on template dependent replication and speci�ccatalysis is considered� Trimolecular �elementrary steps� of simple replicator dynamics are re�solved into two consecutive irreversible reactions� The extreme cases� competition for commonresources and hypercycle�like cooperative feedback� were analyzed in some detail� Although thedynamics of the extended networks resembles corresponding replicator dynamics in general� thereare signi�cant di�erences� Most notably� the interior �xed points in the cooperative model turnedto be are asymptotically stable for an arbitrary number of species whereas simple replicator dy�namics predict an asymptotically stable �xed for four species and less and a stable periodic orbitfor all other cases�

�� Introduction

Spiegelmans pioneering studies �� on replication of Q� RNA and evolution in

vitro initiated a great number of investigations dealing with the design of self�

replicating chemical and biochemical systems� Non�natural molecules were syn�

thesized that can act as templates and undergo self�replication in enzyme�free

environments ��� ��� ��� ��� �� � Catalytic activities of RNA include also limited

capacities for ligation and replication �� � � These results encouraged the sugges�

tion that �early life� consisted of a world of molecular replicators� RNA molecules

or simpler precursors� The existence of an RNA world thus is a promising hy�

pothesis in this context �� � Whether or not a scenario of competing molecular

replicators has played an fundamental role in prebiotic chemistry� the behavior

of collections of interacting replicator species is basic to any theory of molecular

evolution�

In parallel to the advances in template chemistry a theory of molecular evolution

based on chemical reaction kinetics and mathematics of dynamical systems has

been developed by Eigen and Schuster �� �� ��� �� � These studies focussed on a

simpli�ed kind of �over�all� kinetics of replication and ignored all �ner stuctural

and mechanistic details� Replicator dynamics is considered as mass action kinet�

ics of sets of autocatalytic �chemical� reactions neglecting most or all reaction

intermediates� In its most condensed form the kinetics of error�free replication of

interacting molecular species is casted into the autocatalytic reaction network

�M� � Ik � Il �� �Ik � Il � �W � k� l � �� � � � � n� �R�

� � �

Page 4: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

where I is the replicator and �M� denotes the energy�rich building material for

replicators� for example the four activated nucleotides in form of nucleoside triphos�

phates ATP� UTP� GTP� and CTP� �W � accounts for the low�energy �waste� that

accompanies the formation of replicators� Since M is assumed to be bu�ered at

constant concentration and W does not show up in the kinetic di�erential equa�

tions for irreversible reactions� they are both put in parentheses� Often the model

is extended by consideration of the uncatalyzed replication reaction� see e�g� ���

�� �

�M� �Xi � �Xi � �W � �R��

The dynamical system that are most commonly associated with these two reaction

schemes are a second order replicator equation of the form

�xk � xk

�� nX

j��

bkjxj �Xi�j

bijxixj

�A

and a �rst order replicator equation of the form

�xk � xk

�ak �

Xi

aixi

for �R� and �R��� respectively� Originally developed as a model of prebiotic evolu�

tion� 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� The mathematical

properties of replicator equations have been the subject of hundreds of research

papers� see �� and the references therein� One of the major advantages of sim�

ple replication kinetics consists in its straightforward extension to mutation that

is understood as a parallel process to correct replication� Both �rst and second

order replication with mutation have been studied extensively� see e�g� �� �� �� �

for the �rst order and ��� �� for the second order case�

Straighforward extensions of the replicator model deal with two di�erent issues�

�i� Second order replicator equations �R� involve trimolecular encounters whichare generally improbable according to collision theory� In order to avoid un�

likely occurring events trimolecular encounters can be resolved in consecutive

bimolecular encounters �for example see �� and this contribution��

� � �

Page 5: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

�ii� Polymerization kinetics involves a number of individual reaction steps that

are repeated for every monomer to be incorporated into the growing polymer

chain� Replication of templates of a thousand monomers consists of several

thousand individual steps� From the point of view of reaction kinetics repli�

cation of polynucleotides is an overwhelmingly complicated process involving

a large number of intermediates � �

It is by no means self�evident that the global features of such a complicated re�

action mechanism can indeed be captured by an over�all replicator dynamics that

treats replication events quasi as elementary steps and encodes all structural and

functional details into a small set of reaction rate constants� A variety of mech�

anisms based on more involved reaction schemes have been investigated� Many

of them reproduce the qualitative behavior of pure replicator networks quite well

�see� e�g�� ��� �� �� A model involving replication and translation has be formu�

lated and studied by computer simulation �� � This system can be viewed� for

example� as a singularly�perturbed replicator equation �� �

In this contribution we introduce and analyse a mechanism that resolves the tri�

molecular reaction of second order replicator equations �R� into two consecutivebimolecular steps �and thus is an analogue of the previously reported extended

hypercycle model �� ��

M �Xi

gi

���� Xi � Si

Xi � Sjcij

���� �Xi

�I�

This reaction scheme involves an intermediate product Si which� for example� can

be interpreted as a speci�cally activated form of the substrate� According to mass

action kinetics the relations gi � � and cij � � are always ful�lled� In addition weshall include the �rst order �uncatalyzed� replication reaction�

M �Xi

ki

���� �Xi � �I ��

The purpose of this contribution is neither to provide a detailed description of

realistic replication kinetics in a contemporary organism nor to simulate real in

vitro evolution experiments� We are interested here in the validity and the limits

of simple replicator dynamics� To this end we consider here a model that is still

� � �

Page 6: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

much simpler than models of RNA replication kinetics devised by Biebricher and

Eigen �� � � but which is su�ciently complicated to show a signi�cant in�uence of

the resolution of �over�all� kinetics into more detailed elementary steps resulting

in deviations from the replicator scenario� On the other hand� our model is simple

enough to allow for an �at least partial� analytical treatment�

The paper is organized as follows� In section � we outline the mathematical model

and describe its limiting cases by means of singular perturbation theory� Sections �

and � contain detailed descriptions of the competitive and the cooperative cases�

respectively� In section � we discuss the relation of our system to other classes

of autocatalytic networks� Mathematical details are compiled in two appendices�

Appendix A compiles the proofs of the lemmas and theorems quoted in the main

text� A brief discussion of the properties of the Jacobian matrix for our system is

the topic of appendix B�

��Model Equations

Assuming mass action kinetics it is straightforward to write the reaction mecha�

nism �I� I �� in form of kinetic equations�

�xi � xi

�� aki �

nXj��

cijsj � �A

�si � giaxi � si

nXj��

xjcij � si �

���

Explicit expressions for substrate concentration a � M and dilution �ux specify

the dynamical boundary conditions� In the case of constant organization �co� we

have a � � �after a suitable transformation of the time axis� and Pi�xi � si� � �

with a suitable normalization of the concentrations� The total concentration c�

is then contained as an implicit multiplicative factor in the second order rate

constants cij � As a consequence we obtain

CO � anX

j��

�kj � gj�xj �nX

j��

�ixidef

���! � ���

� � �

Page 7: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

Here we de�ne� �idef

��� ki�gi� Note that the form of the �ux function CO is quite

di�erent from the quadratic form in the second order replicator equation� Since

the overall concentration is held constant� the phase space is the unit simplex

S�n �

��x�� � � � � xn� s�� � � � � sn�

�� xi� si � � � X�xi � si� � �

��

In the CSTR setting the concentration of the substrate varies and we have to take

into account its in�uence explicitly� Then� CSTR � r is a constant �ux rate�

while the building material a satis�es the di�erential equation�

�a � a�r � a

�� nX

j��

�ixi � r

�A � ���

For convenience we use the same notation for variables as in the constant organi�

zation setting since most of the results are very similar� As usual in the CSTR�

the total concentration c� � a �P

i�xi � si� converges to a�� Therefore all �xed

points lie on the �n� ��dimensional simplex

S�n���a�� �

��x�� � � � � xn� s�� � � � � sn� a�

��xi� si� a � � � a�X�xi � si� � a�

��

From equ������ we see that if the replicating species Xi vanishes at a certain �xed

point� then the corresponding intermediate Si must also vanish and vice versa�

We can therefore characterize a �xed point PI by its index set I� i�e�� by the set of

the species with positive concentrations at the �xed point� The number of these

species is denoted by jIj� It will be useful therefore the introduce the followingde�nition�

De�nition� An index set I is admissible if the corresponding �xed point

PI exists in a non�empty range of total concentrations or �ux rates in

the CSTR setting� respectively�

The concentrations of the individual species are coupled only via ! or a� respec�

tively� Thus the computation of the equilibrium concentrations is the same for �xed

points in the interior and on the boundary of the simplex because the boundary

consists of subsimplices on which the dynamics is given by a lower dimensional

� � �

Page 8: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

analogue of equation ���� The stability of the �xed points� however� depends of

course on all coordinates not only on those that belong to non�vanishing species�

Let �"x� "s� or �"x� "s� "a� be a �xed point with admissible index set I� Then we can

arrange the Jacobian matrix in the following form

J �

�BBBBB�

J� � � � � �� J� ����

� � �

� � � � � Jn�jIj

X JI

�CCCCCA

where Ji is �� � matrix belonging to a species i �� I� The derivatives with respect

to a and derivatives of �a form the last row and the last column� respectively� in

the CSTR case� The blocks Ji are of the form

Jidef

���

� �xi

�xi

� �si�xi

� �xi

�si� �si�si

aki � �Cs�i � �

agi ��xC�i �

in both the CSTR and the constant organization model� By X we denote a matrix

of expressions that have no in�uence on the eigenvalues of the Jacobian� JI � �nally�

describes the linearized ODE on the subsimplex of the non�vanishing species�

The eigenvalues of the Jacobian matrix of ����� or ����� can therefore by subdivided

into three classes�

�i� Transversal eigenvalues belong to the species that vanish on boundary points�

These are the diagonal entries of the the ��� blocks Ji with i �� I� One of them

is always negative� The form of the other one is reminiscent of the transver�

sal eigenvalues of the second order replicator equation �� � The transversal

eigenvalues describe the stability of a �xed point on the boundary against the

invasion of an additional species�

�ii� The external eigenvalue �"! or �r� respectively corresponds to a left eigen�vector with entries of unity for i � I and zero for l �� I� It decribes the

approach of the system to the simplex S�n���a�� in the CSTR� Under con�

stant organization it re�ects the normalization constraint and has no physical

interpretation at all�

�iii� All other eigenvalues are called internal� They depend only on the non�

vanishing species j � I� that is on the block JI � In contrast to transversal and

� � �

Page 9: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

external eigenvalues it is not possible in general to obtain explicit expressions

for the internal eigenvalues�

Not much can be said about the general model� A few partial results on low�

dimensional equilibria can be found in �� � Two opposite extreme cases� however�

can be analysed with considerable detail�

The competitive case� in which each species replicates independent of all the

other ones can be viewed as a generalization of the Schl�ogl model �� � The in�

teraction matrix C has the entries cij � fi�ij � where �ij is Kroneckers symbol�

The species interact only via substrate and �ux� i�e�� they compete for a common

resource�

In the cooperative case speci�c interactions between di�erent species are a pre�

requisite for replication� We consider here only the most prominent example� Xi

produces Si� which in turn produces Xi�� for i � �� � � � � n � � and and Sn pro�

duces X�� This type of cyclic catalysis is known as the hypercycle �� ��� �� � The

interaction matrix is of the form cij � fi�i���j with the indices taken modulo n�

Before we return to the description of the two�step replication mechanism �I� I ��we consider a few limiting cases� A good introduction to singular perturbation

theory is for instance �� � We shall see that our model �reduces� to replicator

equations in certain singular�perturbation limits� The results discussed in ��� ���

��� �� explain in what sense a singularly perturbed dynamical system of the form

�x � f�x� y� q� ��

� �y � g�x� y� q� ���A�

can be derived from the limiting case �� �� Here x � X and y � Y are the slow

and fast variables� respectively� A � IRp is the admissible set of parameters q and

K X�Y is compact such that intK is simply connected� We will be interested

only in the dynamics of �A� in the compact set K� Suppose the vector �eld g has

the following properties�

�i� There exists a unique smooth function � X � A � Y which solves the

algebraic equation g�x� �x� q�� q� �� � ��

�ii� The Jacobian J�x� q� �g

y�x� �x� q�� q� �� is uniformly stable on X�A� i�e��

there is a positive constant � � � such that real value of all eigenvalues of

J�x� q� is bounded above by �� for all x � X and all q � A�

� � �

Page 10: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

�iii� For �xed q � A we have f�x� �x� q��jx � Xg K �� ��

Then there is an open sets A� � A� and an open interval I � � I such that for all

�q� �� � A� � I � there is a unique integral manifoldMq�� � fy � ��x� q� ��jx � Xgwith the following properties�

�i� � � X � A� � I � � Y is continuously di�erentiable�

�ii� � satis�es lim���

��x� q� �� � �x� q� and lim���

x�x� q� �� �

x�x� q� uniformly

on X � A��

�iii� the long�time behavior of a trajectory passing through a point x� in a suit�

ably small neighborhood of the integral manifoldMq�� is determined by the

dynamics on this manifold� that is� by the di�erential equation

�x � f�x� ��x� q� ��� q� �� �

The limit of the above di�erential equation for �� � is known as the degen�

erate system

�x � F �x� q� def��� f�x� ��x� q� ��� q� �� � �B�

If f is continuously di�erentiable with uniformly bounded derivatives on

X � Y � then there is a function ���� with ���� � � such that

sup fk#�x� q� ��k � k#�xkg� �����

In other words the dynamics of trajectories near the integral manifold is described

by the di�erential equation �x � F �x� q� � #�x� q� �� where #�x� q� �� is a regular

perturbation of the degenerate system �x � F �x� q�� In such a case we say the

singular perturbation problem �A� reduces to the degenerate problem �B� on the

slow manifold for �� �� If �A� reduces to its degenerate system� then the following

propositions are true�

�i� It the degenerate system has a hyperbolic equilibrium $x�� then there is a

hyperbolic equilibrium $x� of �A� nearby� at least for su�ciently small �� In

particular� $x� is asymptotically stable i� $x� is asymptotically stable �� �

�ii� If the degenerate system has a non�degenerate closed orbit $�� with primitive

period T�� then �A� contains a nearby non�degenerate closed orbit $�� with

primitive period T� close to T� for small enough � � � � �

� � �

Page 11: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

�iii� The existence of a transversal homoclinic orbit in the degenerate system im�

plies the existence of a nearby transversal homoclinic orbit in �A� for su��

ciently small � � � ��� Theorem ��� �

We return now to the two�step replication mechanism �I� I ���

Theorem �� The dynamical system ����� in the CSTR setting reduces to the

inhomogeneous replicator equation

�yi � yi

��ki �X

j

cijyj �Xl

yl

��kl �X

j

cljyj

�A �

when the formation of the intermediates is fast� i�e�� for gi ���

Interestingly� the CSTR version of this model does not reduce to the analogous

constant organization equation in the limit r� �� In stead we have

Theorem �� The dynamical system ����� in the CSTR reduces in the limit of

small �ux rates� r � �� to the non�linear replicator equation

�yi � yi

�� fi�y��X

j

yjfj�y�

� �C�

with the response function

fi�y� � ki �

Pj cijgjyjPj cijyj

� �D�

Another interesting limit is obtained when the replication step is very fast� i�e�� if

cij � dij�� and �� �� In the constant organization case this correspond to large

total concentrations c� ��� Interestingly� we obtain the same limit as above inthis case�

Theorem �� The dynamical system ��� in both the constant organization and

the CSTR setting reduces to the nonlinear replicator equation �C� with response

function �D� in the limit large rate constants cij �

The replicator equation �C�D� will be discussed elsewhere� We remark here only

that for all gi � g we have a �rst order replicator equation with �tness values

ki � g�

� � �

Page 12: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

�� Competition

Both in the CSTR and in the constant organization setting we can perform a

fairly complete analysis of the �xed points� Since the criteria for the existence

of �xed points are very similar for both boundary conditions we shall discuss

them together� Except for section ��� we shall assume that all parameters ki are

di�erent� Since the diagonal form of the coupling matrix C is not changed if we

relabel the species� we may assume that that the constants ki form a decreasing

sequence�

The di�erential equations describing the competitive model under constant organ�

isation are��xi � xi�ki � fic�si � !��si � gixi � si�fic�xi �!�

���

For the CSTR model we obtain

�xi � xi�kia� fisi � r�

�si � giaxi � si�fixi � r�

�a � a�r � a�r �Xi

�ki � gi�xi�

���

����Existence of Equilibria

The equilibrium concentrations at the �xed point PI �denoted by bars� are

"si �"!� kific�

"xi �"!�"!� ki�

fic���i � "!� ���

for all i � I in the case of constant organization� For the CSTR we �nd the

equilibrium concentrations

"si �r � ki"a

fi� "xi �

r�r � ki"a�

fi��i"a� r����

In addition there is the trivial �xed point at which all replicating species and

intermediates vanish� "a � a�� It corresponds to the index set I � ��

� �� �

Page 13: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

Lemma �� A �xed point of type PI exists if and only if ki � "! � �i or if

ki � r�"a � �i for all i � I� respectively�

Proof� Obvious�

The extremal indices m�M � I are de�ned by �m � mini�I �i and kM �

maxi�Iki� respectively� For all admissible index sets I �� � we set I � � I n fMg�

Lemma �� �i� If I �� � is admissible then I � is also admissible��ii� I �� � is admissible if and only if kM � �m�

�iii� � is admissible for the CSTR�

Proof� This is a by�product of the proofs of the following theorems�

We shall use the following notation for bifurcation points� A transcritical bifur�

cation involving �xed points of types PI and PJ will be denoted by I � J � Asaddle node bifurcation involving two �xed points of the same type will we indi�

cated by �I� and Hopf bifurcations will be denoted by I�� The symbols are usedas superscripts of the bifurcation parameter for the critical values�

Theorem �� Let I be an admissible index for the competitive model under

constant organization� Then the following assertions are true�

�i� If jIj � � then PI � Sn for all c���ii� If jIj � � then there is exactly one �xed point of type PI � Sn for su�ciently

large values of c�� At a certain total concentration cI�I�

� a transcritical bifur�

cation with the �xed point of type PI� takes place� For c� � cI�I�

� we have

PI �� Sn�

Theorem � Let I be an admissible index for the competitive model in the CSTR�

Then the following assertions are true�

�i� For su�ciently small �ux rates r there is exactly one �xed point of type PI �

�ii� There exists a �nite �ux rate rI�I�

� � at which a transcritical bifurcation

occurs which involves a �xed point PI and a �xed point PI� �

�iii� If r � rI�I�

one of the following three alternatives is true�

�a� There is no �xed point of type PI in the simplex�

�b� There are two �xed points of type PI with di�erent concentrations "a of

the substrate �Figure �a�� In this case we observe a saddle node bifurcation

� �� �

Page 14: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

at a critical �ux rate r�I � rI�I�

�c� There is a single �xed point of type PI in the simplex �Figure �b�� Fur�

thermore we �nd case �a� for all r � r�I and case �b� for all �ux rates r in

the range r�I � r � rI�I�

Two characteristic scenarios for two species and di�erent choices of the rate con�

stants are shown in Figure ��

0.0 0.5 1.0 1.5r

0.0

0.4

0.7

1.1

1.4

1.8

2.2

a

0.0 0.2 0.4r

-0.3

-0.1

0.1

0.3

0.5

a

a b

Figure �� Bifurcation patterns in the competitive case�Stationary concentrations of the substrate �a are shown as a function of the �ow rate rin the CSTR� The parameters are n � �� k� � ���� k� � �� g� � � g� � �� a� � � forboth cases and a� f� � �� f� � �� b� f� � ��� f� � � � Physically acceptable solutionsare con�ned to the region between the two dashed lines determined by equation ����

���� Stability of Equilibria

Under constant organization the corresponding eigenvalues are �"! and kl � "!�

respectively� In the CSTR we have the transversal eigenvalues �r and kl"a � r�

respectively� We shall use the abbreviation I l def

��� I � flg� Three cases can bedistinguished for the transversal eigenvalues�

� �� �

Page 15: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

��� kl � kM� The eigenvalues are negative for all values of c� because "! � kM �or

r � kM"a� respectively� the eigenvalue is always negative both in the CSTR

and under constant organization

��� kl � �m� There is one positive and one negative transversal eigenvalue for

all total concentrations� �The �xed point P Il does not exist� since I lis not

admissible� This is true for both the CSTR and constant organization�

��� kM � kl � �m� Under constant organisation the behavior of the �xed points

is fairly simple� There is a transcritical bifurcation with of PI with PIl at

cI�Il

� �P

i�I �i�kl�ki��fi��i�kl�� For total concentrations below this value�

one of the transversal eigenvalues is positive and one negative� Above this

value� both eigenvalues are negative�

The situation is much more involved in the case of the CSTR since there may

be more than one �xed point with index set I� The behavior of the transversal

eigenvalue can therefore be summarized as follows� For more details and proofs

we refer to �� � At rI�Il

there is a transcritical bifurcation of PIl with �one

of the� �xed point�s� PI � If PIl passes through P���I � its transversal eigenvalue

is positive for �ux rates above rI�Il

� Since "a��� � "a���� it is clear that the

transversal eigenvalue is also positive for P���I � Reverting this argument we

see that if PIl passes through P���I � the transversal eigenvalue is negative for

both �xed points PI at �ux rates above rI�Il � while it is positive for P

���I for

lower �ux rates�

Not much can be said about the internal eigevalues� As far as we could see�

there is little di�erence in the behavior of the model between CSTR and constant

organisation� Explicit expressions are available only for jIj � �� Under constantorganization there is one internal eigenvalue� which is negative� In the CSTR case

we �nd two internal eigenvalues for jIj � � for each of the two �xed points of typePI � for P

���I both are negative� for P

���I one is positive and one negative�

For the case of constant organization it can be shown that the determinant of the

internal part of the Jacobian is negative for even jIj and positive for odd jIj� see�� � Consequently� there must be at least one positive internal eigenvalue for even

jIj� Numerical studies for both boundary conditions indicate that there are alwaysat least jIj � � positive internal eigenvalues� i�e�� only �xed points with jIj � � canbe stable�

� �� �

Page 16: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

For constant organization we have� A �xed point Pfig is stable if �i � kl for all

l and if c� � cI�Ij

� for all j with kj � ki� Numerical computations suggest that

these are the only stable equilibria� The corresponding result for the CSTR is� A

�xed point P���fig is stable if kl � �i for all l and r � rI�I

j

for all j with kj � ki�

Numerical simulations again suggest that all other �xed points are unstable�

���� Special Case The Symmetric Model

A more detailed analysis is possible in the special case of equal reaction constants

for all species� i�e�� if ki � k� gi � g� and fi � f for i � �� � � � � n� The symmetry

in the rate constants induces symmetry in the equilibrium concentrations� "xi � "x

and "si � "s for all i � I� Under these conditions a fairly complete stability analysis

of the �xed points is possible� The properties of the Jacobian are discussed in

appendix B�

Let us consider the constant organization model �rst� The di�erential equations

simplify to�xi � xi�k � c�fsi � !��si � gxi � si�c�fxi �!�

���

where � � k � g and the �ux is given by ! � �P

xi� The equilibrium concentra�

tions at the �xed point PI are

"xi �"!�"!� k�

fc���� "!� � "si �"!� k

c�f���

for all i � I� Since � � k� all nonempty index sets are admissible� Thus we have

�n � � �xed points� The �ux can also be obtained explicitly for all �xed points�

"! ���jIjk � c�f�

jIj�� c�f����

and we �nally obtain the coordinates of the �xed points in terms of the kinetic

constants only�

"xi ��

jIj �jIjk � c�f

jIj�� c�f� "si �

g

jIj�� c�f����

� �� �

Page 17: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

for all i � I� Note that symmetry implies xi � si � ��jIj for all i � I� All �xed

points exist for all values of c�� in contrast to the general model� where only �xed

points with jIj � � exist for all total concentrations� there are no transcritical

bifurcations in the symmetric case�

The di�erential equations for the CSTR case simplify to

�xi � xi�ka� fsi � r�

�si � gxia� si�fxi � r�

�a � a�r � a

�r � �

Xi

xi

� ����

All possible index sets are admissible� i�e�� there are �n admissible index sets� The

equilibrium concentrations at a �xed point PI are

"x �r�r � k"a�

f��"a� r�� "s �

r � k"a

f� ����

The equilibrium concentration of the building material ful�lls�

"a���f � jIjk�� "a�fa��� fr � jIj�r� � fa�r � � � ����

Thus there is a transcritical bifurcation with the trivial �xed point at rI�� def

��� ka�for all index sets� We have to distinguish two cases� If f � f c � jIjk� � g � jIjkthen there is a saddle node bifurcation at

r�I �a��f

�pf � jIjk �pjIjg�� � a�I �

a�f

f � jIjk �p�f � jIjk�jIjg � ����

Otherwise there are no �xed points of type PI in the simplex for r � rI���

Exploiting the symmetry of the �internal part of the� Jacobian� we can calculate

all internal eigenvalues� see appendix B� We obtain essentially the same result for

both boundary conditions�

Theorem �� All �xed points with jIj � � are stable and all remaining �xed pointsare unstable in the case of constant organisation� All �xed points with jIj � � areunstable in the CSTR case� A �xed point PI with jIj � � is stable if and only ifit is either the only one of this type in the simplex or if has the lower substrate

concentration in the case that there are two such �xed points�

The phase portrait of the symmetric case is shown in Figure �� It is in complete

agreement with the corresponding trajectories of the simple replicator system�

� �� �

Page 18: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

���� Summary

We �nd close similarities between the CSTR and the constant organization model�

Appart from details we observe the same behavior when c� is increased or r is

decreased� respectively� This does not come as a surprise� For the replicator

equation it was shown recently �� that the CSTR can be viewed as a singularly

perturbed constant organization model in the limit r � �� In this limit the full

capacity of the CSTR is �lled with polymeric materials� i�e�� c� �P

i�xi � si�

approaches a�� We can therefore interpret c� and ��r as conceptually the same

bifurcation parameter�

y1

y3

y2

Figure �� The competitive model with n� under constant organization�Constants for di�erent species are chosen to be equal� Trajectories are shown for sumsof concentrations� yi � xi� si� The parameters used are� k � ��� c� � � f � ��� andg � ��

Secondly we �nd a cascade of transcritical bifurcations as c� or ��r increases� in

which �xed points with an increasing number of species are introduced into the

� �� �

Page 19: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

physically meaningful part of the state space� This is due to the fact that c� or

r a�ect the relative importance of the �rst order and the second order reactions�

For very small c� the �rst order term dominates which leads to the selection of a

single species� i�e�� there is a single stable �xed point� In the CSTR only the trivial

�xed point is stable if r becomes very large� of course�

For larger c� �or small r� the model behaves just like a replicator model� all ��

species equilibria are stable� all other �xed points are unstable� This matches

exactly the phase portrait of the Schl�ogl model �� in the replicator equation

setting� We conclude hence� that the competitive model with intermediates in

essence reproduces the behavior of the corresponding inhomogeneous replicator

equation �� �

�� Cooperation

The di�erential equations for the cooperative model under constant organization

are�xi � xi�ki � fi��c�si�� � !��si � gixi � si�fic�xi�� � !�

����

with the �ux term ! �nX

j��

�ixi� The total concentration c� will again be used as

a bifurcation parameter in the constant organization case� The kinetic equations

for the CSTR model read

�xi � xi�kia� fi��si�� � r�

�si � gixia� si�fixi�� � r�

�a � a�r � a�r �nX

j��

�kj � gi�xj�

����

Unless stated otherwise� we assume that the constants ki are all di�erent� Without

loosing generality we can then label the species so that k� is the largest rate

constant of a �rst order reaction�

Due to the form of the interaction between the reactants� there is a sharp contrast

between the �xed point�s� in the interior and those on the boundary of the simplex

� �� �

Page 20: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

in the mutualistic model� While in the former case we have a closed catalytic

cycle� in the latter there are only catalytic chains� We can show that all �xed

points on the boundary are unstable if there is an interior �xed point� If we omit

the uncatalyzed formation �ki � ��� there are no �xed points on the boundary of

the simplex�

����Boundary Equilibria

The non�empty admissible index sets are the same for constant organization and

the CSTR� The set N � f�� � � � � ng belongs to the interior rest point�s��

Lemma �� Let I �� � be admissible� Then�i� there are indices j and k such that I � fj� j � �� � � � � k � �� kg� i�e�� the speciesin a boundary �xed point form a catalytic chain�

�ii� kj � maxi�I ki�

�iii� The set I � � I n fkg is admissible��iv� If k� � k� � � � � � kn then there are n�n � ���� admissible set� The is the

largest possible number�

�v� If k� � kn � kn�� � � � � � k� then there are only �n� � admissible sets� whichis the smallest possible number�

Theorem �� Let I be an admissible index set� Then there exist constants cI�I�

and rI�I�

� a�kj for the constant organization case and the CSTR with jIj � ��

respectively such that�

�i� There is a unique �xed point of type PI in the physically meaningful part of

the state space if c� � cI�I�

� or r � rI�I�

� respectively�

�ii� A transcritical bifurcation I � I � occurs at the critical value��iii� There is no �xed point of type PI in the simplex if c� � cI�I

� or r � rI�I�

respectively�

In addition we have the trivial �xed point P� at which only the substrate is present

in the CSTR� "a � a�� It is easy to check that it is stable if and only if r � a�k��

i�e�� if r�a� exceeds the largest of the kinetic constants ki�

� �� �

Page 21: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

Let us now turn to the stability of the non�trivial boundary equilibria� In com�

plete analogy to the discussion of the competitive model we distinguish external�

transversal� and internal eigenvalues� No useful expressions were obtained for the

internal eigenvalues except for equilibria with jIj � �� Explicit expressions for thetransversal eigenvalues are compiled in table �� It will be convenient to use the

notation $I def��� I � fk � �g� i�e�� $I � � I�

Table ��Transversal eigenvalues of boundary equilibria

eigenvalues number signconstant organization CSTR

�kj �r n� jIj � � ���kj � ks� �r�kj � ks� n� jIj � � �

��kj � c�fk��"xk� ��r � fk��"xk� � ���kj � kk��� � c�fk"sk ��r � kk��"a� � fk"sk � y

� For all s �� �I� There is at least one s such that ks � kj unless j � �� i�e�� at least one of theseeigenvalues is positive if j �� ��

y Negative for small enough values of c� or large enough values of r� respectively� providedkj � kk��� i�e�� if �I is admissible�

A boundary equilibrium is saturated ��� �� if all transversal eigenvalues are non�

positive� An immediate consequence of the expressions in table � is the following

result�

Lemma �� If PI is a saturated boundary equilibrium then � � I�

If PJ � J �� N is contained in the physical state space then no �xed point PI with

I J can be saturated� In the case of constant organization this is also true for

J � N �

In the constant organization model we have the following bifurcation pattern� For

su�ciently small values of c� there are the n boundary equilibria Pfig correspond�

ing to isolated species� Of these Pf�g is stable �and we suspect globally stable��

As c� increases Pf���g passes through Pf�g in a transcritical bifurcation and be�

comes stable in an interval bounded below by cf�g�f���g� � If c� increases further�

however� there might be a Hopf bifurcation that makes Pf���g unstable before the

transcritical bifurcation with Pf����g� In fact� there is a cascade of bifurcations of

� �� �

Page 22: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

the type I � $I as c� increases until� at su�ciently high values of c�� there exists aninterior equilibrium PN � At this point all boundary equilibria have become unsta�

ble� Analogous cascades of bifurcations may exists for the �xed points belonging

to other admissible sets� say f�g � f�� �g� etc� These families involve however onlyunstable equilibria�

We �nd almost the same behavior in the CSTR when r is reduced� For su�ciently

large �ux r the trivial �xed point is globally stable� If r decreases below r��f�g then

Pf�g becomes stable� and we observe the same cascade of transcritical bifurctions

as above when r is decreased further� Only the �nal step can be more complicated�

The interior �xed point either enters the simplex via the transcritical bifurcation

N �N � �Figure �b�� or there is a saddle node bifurcation N� which gives birth toa pair of interior �xed points one of which leaves the simplex through N � N � at

an even lower �ux rate �Figure �a��

0.0 0.5 1.0 1.5r

-1.0

0.0

1.0

2.0

a

0.0 0.1 0.3 0.4 0.5 0.7 0.8r

-0.5

0.0

0.5

a

a b

Figure �� Bifurcation diagrams for the cooperative case�Stationary concentrations of the substrate �a are shown as a function of the �ow rate rin the CSTR� The parameters are n � �� k� � ���� k� � �� g� � � g� � �� a� � � forboth cases and a� f� � �� f� � �� b� f� � ��� f� � � � Physically acceptable solutionsare con�ned to the region between a � and the dashed line�

� �� �

Page 23: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

���� Interior Equilibria

It is not possible to obtain complete analytical results on the stability of the

interior equilibria� Their number and conditions for their existence� however� can

be determined explicitly� For the constant organization model the situation is

quite simple� We can rule out the occurrence of bifurcations that would change

the number of rest points in the interior of the state space while bifurcations of

the boundary can be treated explicitly�

Theorem � Let PN be an interior equilibrium of the cooperative model under

constant organization� Then its Jacobian is regular� i�e�� all eigenvalues are non�

zero�

Theorem �� There is a critical value cN��N

� � � such that there is no interior

rest point for all c� � cN��N

� � � and there is unique interior rest point for all

c� � cN��N

� � � in the cooperative model under constant organization�

The situation is more involved for the CSTR� We have

Theorem ��� There are at most two interior equilibria in the cooperative model

for the CSTR� In more detail we have�

�i� For su�ciently small �ux rates� there is exactly one interior �xed point� At

r � rN�N �

there is a transcritical bifurcation of an interior �xed point with

the unique �xed point PN � �

�ii� There are two ways in which this bifurcation can happen�

�a� The �xed point that lies inside the simplex may pass through PN � and

leaves the simplex� In this case there is no interior rest point for r � rN�N �

�b� a second �xed point enters the simplex� In this case there is a saddle

node bifurcation which annihilates the two interior equilibria at a �ux rate

r�N � rN�N �

% there is no interior �xed for any r � r�N �

� �� �

Page 24: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

���� Special Case The Symmetric Model

A fairly complete stability analysis can be performed if we set ki � k� gi � g� and

fi � f � We have�xi � xi�k � c� f si�� � !��si � gxi � si�c� f xi�� �!�

����

with ! � �P

xi in the case of constant organization and

�xi � xi�ka� fsi�� � r�

�si � gxia� si�fx� r�

�a � a�r � a�r � �Xi

xi�

����

for the CSTR� It is easy to characterize the admissible index sets�

Lemma �The set N is admissible� A nonempty set I �� N is admissible if and

only if it does not contain two indices i and j with ji� jj � � �where the indicesare taken modulo n� In addition� the empty set is admissible in the CSTR setting�

Lemma �� All boundary equilibria �except the trivial one� are unstable both

under constant organization and for the CSTR�

The elementary hypercycle �xi � xi�xi�� � !� has the same properties ��� �� �

Theorem ��� The symmetric cooperative model under constant organization has

a unique interior equilibrium for all c� � �� This �xed point is stable if c� is large

enough�

�i� For n � � the interior equilibrium is always stable�

�ii� The interior �xed point is stable for both small and large value of c� for n � ��

If g � �����k then there are two Hopf bifurcations with � � c��� � c��� � and

the interior rest point is unstable c��� � c� � c��� �

�iii� For n � � we have a single Hopfbifurcation at c�� � ��g � k��f if g � k� In

this case the equilibrium is unstable for � � c� � c��� If g � k then the interior

rest point is stable for all c� � ��

�iv� For n � � there is a single Hopfbifurcation at c�� � �� The interior equilibrium

is unstable for c� � c�� and stable for c� � c���

� �� �

Page 25: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

Again the situation is more involved for the CSTR�

Lemma �� It r � a�k there is a unique interior rest point� There is no interior

rest point for r � a�k if f � f c � nk��g� If f � f c there are two interior rest

points for

a�k � r � r� �a��f

�pf � jIjk �png�� � ����

and there are no interior rest points for r � r��

0.024 0.026 0.028 0.030x1

0.024

0.026

0.028

0.030

x 2

Figure �� The cooperative model with n� under constant organization�The constants for di�erent species are chosen to be equal� k � ��� f � ���� g � ���and c� ranges from ������ in �� steps from left to right�

Let us now turn to the stability of the interior equilibria�

Theorem ��� For the symmetric cooperative model in the CSTR setting the

follwing is true�

�i� If there are two interior equilibria then at most one of them is stable�

� �� �

Page 26: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

�ii� For small enough �ux rates there is a unique interior equilibrium which is

asymptotically stable�

�iii� The trival �xed point is stable for all r � a�k�

�iv� If n � � then P���N is stable whenever there is an interior rest point% P

���N is

unstable whenever it exists�

�v� Suppose f � f c� i�e�� there is at most one interior rest point�

For n � � there are two Hopf bifurcation is g�k is su�ciently large� The �xed

point is stable for both small and large �ux rates and unstable in between

�Figure ��� If g�k is small there is no Hopf bifurcation and the interior rest

point is always stable�

For n � � there is a single Hopf bifurcation that leads from a stable �xed

point to a saddle if g � k� Otherwise the �xed point is always stable�

For n � � there is always a Hopf bifurcation�

�vi� If f � f c� i�e�� if there can be two interior rest points� it depends on the value

of f � at which of the two �xed points the Hopf bifurcation�s� take�s� place�

Generally speaking� the Hopf bifurcations occur at P���N for small f and at

P���N for large f � In the latter case P

���N is stable at all �ux rates at which it

exists�

���� Summary

As in the competitive model there is little di�erence in the qualitative behavior

of CSTR and constant organization� A cascade of transcritical bifurcations intro�

duces �xed points with an increasing number of species into the system as c� and

��r increase� These are either stable or associated with stable limit cycles� This

phenomenon has been observed is a variety of quite di�erent dynamical systems

that describe replication and selection� as the capacity c� of the environment in�

creases the number of species that can be sustained increases as well� see e�g�� ���

��� ��� ��� �� �

The second order replicator equation corresponding to the limit of larger c� or

small r� respectively� is the elementary hypercycle �� � Our model exhibits a

� �� �

Page 27: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

single� stable interior �xed point in this limit� The elementary hypercycle� on the

other hand� shows stable limit cycles for n � � �� � In our present model we �nd

stable limit cycles also for n � � and n � ��

��Discussion

Most of the previous work on autocatalytic networks was based on the trimolecular

�over�all� mechanismM�I�C � �I�C for the replication of I in the presence of

a catalyst C� The corresponding dynamical systems� replicator equations� Lotka�

Volterra equations� and to a lesser extent the equations for the reactions in the

CSTR are well understood� see �� � This family of dynamical systems is properly

characterized as being replicator�like� The models are su�ciently simply to allow

for a very detailed analysis with analytical methods� On the other hand� a single�

step �over�all� replication kinetics is certainly a very restrictive assumption� It has

been argued repeatedly that this is a good approximation but only a few papers

tried to substantiate this claim� see e�g� ��� ��� ��� ��� �� �

The range of validity of replicator�like models determines whether these systems

can provide a useful description of the dynamics of prebiotic evolution scenarios or

contemporary populations� This calls for a novel type of perturbation theory that

deals with qualitative modi�cations of the model rather than with mere changes

of the reaction constants� As a �rst step we consider here the dynamics of a

model in which the trimolecular replication step is resolved into two irreversible

bimolecular steps� �i� the formation of an �activated� intermediate and �ii� the

template directed conversion of this intermediate into a copy of the template�

The simple observation that the �ux term ! is a linear instead of a quadratic

function indicates that this model is only a rather distant relative of the second

order replicator equations� Nevertheless the model reduces to replicator equations

in certain limits� Furthermore we observe a number of analogies to second order

replicator equations in a very broad range of parameters� Both descriptions of

autocatalytic networks exhibit very similar dynamics irrespective of whether the

reactions are considered under constant organization or embedded in a CSTR� The

reactor capacity c� �in the constant organization case� plays the role of the inverse

� �� �

Page 28: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

�ux rate ��r �in the CSTR�� Using c� or ��r� respectively� leads to very similar

bifurcation diagrams� For small c� uncatalyzed replication dominates� we observe

survival of the �ttest� i�e�� of the species with largest �rst order rate constant�

For large c� the catalyzed reactions determine the behavior of the system� In

the competitive case we observe selection of the most frequent species� quite as

in Schl�ogls classical model �� � For the cyclic arrangement of catalytic coupling�

corresponding to the hypercycle� we observe stable coexistence of all species� The

details of the dynamics are di�erent� however� While the elementary hypercycle

always leads to stable limit cycles for more than � species� we �nd a single stable

�xed point in certain parameter ranges of the two�step model�

We conclude hence� that resolving the replication step into a two�step mechanism

does not lead to drastic changes� While details of the dynamics are di�erent� the

qualitative picture is still essentially the same as in the second order replicator

equation� This paper is considered as a �rst step towards answering the question

that motivated this work� What are the modi�cations to a reaction scheme that

leave global dynamics intact & and which changes will lead to a dramatic alter�

ations of the dynamical behavior' More variations of the replicator theme still

need to be conceived and analyzed before a �nal answer can be given�

Acknowledgements

This work was supported in part by the Austrian Fonds zur F�orderung der Wissen�

schaftlichen Forschung� Project Nos� P�����CHE and P������CHE�

References

� D� Ansov� On limit cycles of systems of di�erential equations with a small

parameter at the highest derivatives� Mat� Sbornik� ����������� �����

� C� Biebricher� M� Eigen� and W� C� Gardiner� jr� Quantitative analysis of se�

lection and mutation in self�replicating RNA� In L� Peliti� editor� Biologically

Inspired Physics� pages �������� New York� ����� Plenum Press�

� �� �

Page 29: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

� C� K� Biebricher and M� Eigen� Kinetics of RNA replication by Q� replicase�

In E� Domingo� J� J� Holland� and P� Ahlquist� editors� RNA Genetics� Vol�I�

RNA Directed Virus Replication� pages ����� CRC Press� Boca Raton� FL�

�����

� J� Doudna� S� Couture� and J� Szostak� A multi�subunit ribozyme that is a

catalyst of and a template for complementary�strand RNA synthesis� Science�

�������������� �����

� J� Doudna� N� Usman� and J� Szostak� Ribozyme�catalyzed primer exten�

sion 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� J� McCaskill� and P� Schuster� The molecular quasispecies � An

abridged account� J� Phys� Chem�� ������������� �����

� M� Eigen� J� McCaskill� and P� Schuster� The molecular quasispecies� Adv�

Chem� Phys�� ����������� �����

� M� Eigen and P� Schuster� The hypercycle� A� Emergence of the hypercycle�

Naturwiss�� ����������� �����

�� M� Eigen and P� Schuster� The hypercycle� B� The abstract hypercycle� Natur�

wiss�� �������� �����

�� M� Eigen and P� Schuster� The hypercycle� C� The realistic hypercycle� Natur�

wiss�� ����������� �����

�� 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� �������� �����

�� I� Epstein� Competitive coexistence of self�reproducing macromolecules� J�

Theor� Biol�� ����������� �����

� �� �

Page 30: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

�� M� Famulok� J� Nowick� and J� Rebek Jr� Self�Replicating Systems�

Act�Chim�Scand�� ���������� �����

�� N� Fenichel� Geometric singular perturbation theory for ordinary di�erential

equations� J� Di�� Eqns�� ��������� �����

�� T� Gard and T� Hallam� Persistence in food webs I� Lotka�Volterra food

chains� Bull� Math� Biol�� ����������� �����

�� B� Gassner and P� Schuster� Model studies on RNA replication I� The

quasiequilibirum assumption and the analysis of a simpli�ed mechanism�

Monatsh�Chem�� ������������ �����

�� R� Gesteland and J� Atkins� editors� The RNA World� Cold Spring Harbor

Laboratory Press� Cold Spring Harbour� NY� USA� �����

�� R� Happel� R� Hecht� and P� F� Stadler� Autocatalytic networks with trans�

lation� Bull� Math� Biol�� ����� in press� SFI preprint ����������

�� R� Happel and P� F� Stadler� Autocatalytic replication in a CSTR and con�

stant organization� J� math� Biol�� ����� submitted� SFIy preprint ����������

�� 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� �����

�� 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 T� G� Hallham� L� J� Gross� and S� A� Lewin� editors� Proceedings

on the Second Autumn Course on Mathematical Ecology� Trieste� Italy� pages

�������� Singapore� ����� World Scienti�c�

�� J� Hofbauer� J� Mallet�Paret� and H� L� Smith� Stable periodic solutions for

the hypercycle system� J� dyn� and di�� equ�� ���������� �����

�� J� Hofbauer and P� Schuster� Dynamics of linear and nonlinear autocatalysis

and competition� In P� Schuster� editor� Stochastic Phenomena and Chaotic

Behavious in Complex Systems� pages �������� Springer�Verlag� Berlin� �����

� �� �

Page 31: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

�� J� Hofbauer� P� Schuster� and K� Sigmund� Competition and cooperation is

catalytic selfreplication� J� Math� Biol�� ����������� �����

�� J� Hofbauer and K� Sigmund� Dynamical Systems and the Theory of Evolution�

Cambridge University Press� Cambridge U�K�� �����

�� H� Knobloch and B� Aulbach� Singular perturbations and integral manifolds�

J�Math�Phys�Sci� ����������� �����

�� J� S� Novick� Q� Feng� and T� Tjivikua� Kinetic studies and modeling of a

self�replicating system� J� Am� Chem� Soc�� ������������� �����

�� L� Orgel� Molecular Replication� Nature� �������� �����

�� J� R�E� OMalley� 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� ������� � ���� �����

�� K� R� Schneider� Singularly perturbed autonomous di�erential systems� In

H� Bothe� W� Ebeling� A� Kurzhanski� and M� Peschel� editors� Dynamical

Systems and Environmental Models� VEB�Verlag� Berlin� �����

�� P� Schuster and K� Sigmund� Replicator dynamics� J�Theor�Biol�� ��������

���� �����

�� P� Schuster� K� Sigmund� and R� Wol�� Dynamical systems under constant

organisation I� Topologigal analysis of a family of non�linear di�erential equa�

tions � a model for catalytic hypercycles� Bull� Math� Biol�� ����������� �����

�� J� So� A note of global stability and bifurcation phenomenon of a Lotka�

Volterra food chain� J� Theor� Biol�� ����������� �����

�� S� Spiegelman� An approach to the experimental analysis of precellular evo�

lution� Quaterly Review of Biophysics� ���������� �����

�� P� F� Stadler� Complementary replication� Math�Biosc�� ����������� �����

� �� �

Page 32: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

�� P� F� Stadler� Dynamics of small autocatalytic reaction network IV� Inhomo�

geneous replicator equations� BioSystems� �������� �����

�� P� F� Stadler� W� Schnabl� C� V� Forst� and P� Schuster� Dynamics

of autocatalytic reaction networks II� Replication� mutation and catalysis�

Bull�Math�Biol�� ��������� �����

�� P� F� Stadler and P� Schuster� Mutation in autocatalytic networks � An anal�

ysis based on perturbation theory� J� math� Biol�� ����������� �����

�� A� Tichonov� Systems of di�erntial equations with a small parameter at the

derivatives� Mat� Sbornik� ����������� �����

�� G� von Kiedrowski� Minimal replicator theory I� Parabolic versus exponential

growth� In Bioorganic Chemistry Frontiers� Volume �� pages �������� Berlin�

Heidelberg� ����� Springer�Verlag�

Appendix A� Proofs of Lemmas and Theorems�

Most of the proofs involve quite tedious manipulations of algebraic equations which

have been performed using the symbolic mathematics packages Mathematica and

Reduce� We present only outlines of some of the proofs and we frequently omit

explicit expressions when they are too complicated to allow for a direct interpreta�

tion� A more complete presentation of this material can be found in the doctoral

thesis �� �

Proof of Theorem �� Let yi �gixiPj gjxj

� Then xi �g��i yiPj g

��j yj

andXj

yj � ��

A short calculation yields

�yi � yi

���ki �

Xj

cijsj �Xl

yl

��kl �X

j

cljsj

�A���

�si � gixi � siXj

�kj � cij � gj�xj

� �� �

Page 33: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

Now we set gi � �i��� The second equation becomes

� �si ��P

l yl��l

���yi � si � �

Xj

si�i�kj � cij�

��� �

In the limes � � � this reduces to si � yi� In order to verify that this solution is

stable we compute the Jacobian

�sisj

� ��ij �

�P

j ���j yj

���X

j

cij���j yj � � � �

Xj

kj���j yj

�A � � �

which is stable for all y� Thus ����� is in fact a stable singular perturbation of the

inhomogeneous replicator equation�

Proof of Theorem �� In order to verify that the reaction vessel is completely

�lled with replicating species and the intermediates we follow the discussion in

�� � We �nd that a�r remains �nite in the limit r � ��

Hence we can use the transformation b � a�r� Furthermore we set ui � si�r and

introduce ! �P

j�kj � gj�xj � This yields

�xi � xi

��ki � �

b

Xj

cijuj � ��A

r �ui � gixi � uib

�� nX

j��

cijxj � r

�A �

r �a � �! � a�b� r

In the limit r � � we obtain the �nite values

ui �bgixia�

a�P

j cijxj�

and b � a��!� This solution is stable since the Jacobian

� �ui

�uj� � �ij

r

�Pj cijxj � r

�� �b�uj

� �

� �ui

�b� �

b�

�Pj cijxj � r

�� �b�b� �a�

b�

� �� �

Page 34: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

has block�diagonal form and all eigenvalues are negative for all x� Thus the equa�

tion for x becomes

�xi � xi

ki �

Pj cijgjxjPl cilxl

� !

a�

in the singular limit� Introducing the new variables yi � xi�P

j xj �nally yields a

non�linear replicator equation with response function fi�y��

Proof of Theorem �� The proof is the same for both the CSTR and the constant

organization case� Set Let cij � dij�� and Di �P

j dij and de�ne ui � Disi���

We obtain

�xi � xi

��aki �X

j

dijujDj

� �A

� �ui � Di

��giaxi � �ui

Di

�� nX

j��

dij�xj �

�A�A

�a � �aXj

�kj � gj�xj � r�a� � a�

In the limit �� � we �nduiDi

�giaxiPj dijxj

because the Jacobian

�uiuj

����x

� ��ijui��X

j

dij�xj � !

�A � �

of the fast variables is stable� Thus we obtain for both constant organization and

the CSTR

�xi � xi

a

ki �

Pj gjdijxjPl dilxl

For the CSTR we have in addition�

�a � �aXj

�kj � gj�xj � r�a� � a� �

Introducing the rescaled variables by yi � xi�P

j xj we arrive at a nonlinear

replicator equation with response function

ki �

Pj dijgjyjPj dijyj

� ki �

Pj cijgjyjPj cijyj

� fi�y�

� �� �

Page 35: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

as in theorem ��

Proof of Theorem �� The equilibrium �ux "! in equ���� can be determined

by the condition that the normalization conditionP

i�si � xi� � �� With the

de�nition

F �!� c��def��� �� �

c�

Xi�I

!� kifi

� �

!

�i � !

an equilibrium is characterized by F �"!� c�� � �� This alegbraic equation has order

sI � � in !� where sI is the number of di�erent values of �i� i � I�

For index sets with jIj � �� i�e� of the form I � fig� we can solve directly for thecoordinates of the �xed point�

"xi �c�fi � kic�fi � �i

� "si �gi

c�fi � �i� "! �

�i�c�fi � ki�

c�fi � �i�

In this case the �xed point lies in the simplex for all total concentrations�

At ! � �i� F �c��!� goes to in�nity and changes sign% there is at least one solution

between each pair of two adjacent lines ! � �i and ! � �j � None of these sI � �solutions lies in the simplex� Furthermore� there is one solution with a negative

value of the equilibrium �ux� This solution is unphysical as well� Consequently

there is at most one solution in the simplex� We observe that F �kM� c�� is negative

for

c� � cI�I�

� �Xi�I

�i�kM � ki�

fi��i � kM�

and positive otherwise� Furthermore� F ��m��� c�� � � for su�ciently small � � ��It follows that there is one �xed point in the simplex for c� � cI�I

� and none for

c� � cI�I�

� � The �xed point PI enters the simplex at cI�I�

� via a transcritical

bifurcation with P �I �

Proof of Theorem � The equilibrium concentration of the substrate is implicitly

given by F �"a�r�� r� � � with the function

F �a� r� def

��� a� � a�Xi�I

r � kia

fi

� �

r

�ia� r

If there are sI di�erent values of � in I� the equation is of order sI � �� We have

not found an analytical solution for arbitrary jIj�

� �� �

Page 36: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

F �a� r� approaches in�nity and changes sign at each of the the lines a � r��i� and

there is at least one solution between each adjacent pair of such lines� These s� �solutions cannot lie in the simplex� Thus there can be at most two solutions on

the simplex� F �a� r� is positive along the line a � r�kM� if

r � rI�I�

�a�kM� � (I

with (I �Xi�I

�i�kM � ki�

fi��i � kM�� �

Since F �r� �r��m� � �� is negative for su�ciently small � � �� there is one solution

if r � rI�I�

� and two or none if r � rI�I�

� At r � rI�I�

there is a transcritical

bifurcation of PI with PI� � For �xed points with jIj � �� this means that Pfig

passes through the trivial �xed point at rfig�� � a�ki�

There are two ways in which this bifurcation can happen� either the interior �xed

point leaves the simplex or a second �xed point enters it� In the latter case the

�xed point that enters the simplex in a transcritical bifurcation has the higher

value of "a% we call it P���I � There is a saddle node bifurcation in which the �xed

points P���I and P

���I disappear at some �ux rate rI� � rI�I

� because neither of

these �xed points can leave the simplex as we raise the �ux rate and because we

know that only the trivial �xed point exists at su�ciently high �ux rates�

Proof of Theorem �� Let us �rst consider constant organization� The eigen�

values of the Jacobian can be computed explicitly by the method described in

Appendix B� We �nd that there is one eigenvalue ��jIj� � c�f� � � and jIj � �pairs of eigenvalues of the form

� ��

�c�f "x� "!�

q�c�f "x� "!�� � �fc�"x��� "!�

It is not hard to check that each pair consists of a positive and a negative eigen�

value� Hence a �xed point with jIj non�vanishing species has jIj�� positive and jIjnegative eigenvalues negative internal eigenvalues as predicted by our conjecture

for the general case� The transversal eigenvalues are �"! and k � "!� They are allnegative since k � "!� It follows that all �xed points with jIj � � are stable andall remaining �xed points are unstable�

The eigenvalues of the Jacobian can be obtained for the CSTR case by the same

method� There are jIj � � pairs of internal eigenvalues of the form

� ��

��f "x� r �

p�f "x� r�� � �f "x��"a� r�

� �� �

Page 37: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

Half of these eigenvalues are positive� the other ones are negative� The explicit

expressions for the remaining two internal eigenvalues are very complicated� They

can be found in �� � It is not hard to show� however� that for the �xed point with

the lower equilibrium concentration of the substrate both are negative� while for

the other �xed point �if it exists� one eigenvalue is positive and the other one is

negative�

Proof of Lemma �� Suppose I contains two distinct indices j and j� with the

property that both j� � and j�� � are not contained in I� Then "sj�� � "sj��� � �implies "xj�kj � !� � "xj��kj� � !� � � contradicting the assumption that all

parameters ki are di�erent� Thus there is only a single index j � I such that

�j � �� �� I� Now we see that "! � kj at the �xed point PI � which in turn implies

the equilibrium conditions "si � r�kj � ki�����c�fi� for all i � I n fkg where k isthe unique index for which �k� �� �� I� Hence kj � ki for all i � I n fjg� i�e�� item�ii� is true� This implies in turn that � � I if and only if j � � and hence item �i�

is true as well� The remaining assertions follow immediately�

Proof of Theorem �� The coordinates of the boundary equilibria can be com�

puted explicity� Under constant organization we �nd�

"si �kj � ki��

c�fiif i � I n fkg

"sk �gk

fk��Pi�I

�i��i�j�k��

��� �

c�

Xi�I

�i

k��Xl�i

��i�j�l

"xi � ki

�"skfk��gk

��i�j�k�� �

c�

k��Xm�i

��i�j�m

�if i � I

��i�j�mdef

����

fm

mYl�i

kj � kl��gl

� �

Lemma � implies that ��i�j�m � � for all i� j and �

�i�j�m � � if i is not the smallest

index in I� The �xed point lies on the simplex if

c� � cI�I�

�def

���

kXi�j

�i

k��Xl�i

��i�j�l � � �

� �� �

Page 38: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

For the CSTR we �nd the equilibrium concentrations

"si �r�kj � ki���

fikjfor i � I n fkg

"sk �

a�kj � r

�� � �j�k �K�i�j� � kj

k��Pi�j

k��Pm�i

��i�j�m

kj

��� � kjfk��

gk

kXi�j

��i�j�k��

�A

"xi � "skkjfk��gk

��i�j�k�� � r

k��Xm�i

��i�j�m for i � I

with the abbreviations

��i�j�mdef

����

fm

mYl�i

kj � kl��gl

� ��j�k� def

���

k��Xi��

kjfi

� K�i�j� def

���

k��Xi��

ki��fi

Thus there is only one �xed point PI for I �� N � It lies on the simplex i�r � r�I�I

�� def

���a�kj�

� � ��j�k� �K�j�k� � kjPk��

i�j

Pk��m�� �

�i�j�m

�because "sk �which is linear in r� becomes negative if this condition is violated�

By construction we have �j�k � K�j�k�� hence the rI�I�

� �� The same argument

implies that the denominator cannot be smaller that � and hence a�kj � rI�I�

Proof of Theorem � In order to show that the Jacobian is regular we demon�

strate explicitly that its rows are linearly independent� i�e�� that there is no non�

trivial solution of the system of linear equationsnX

i��

�i �xlxi

�nX

i��

�i �xlsi

� �

nXi��

�i �slxi

�nX

i��

�i �slsi

� �

with variables �i and �i� � � i � n� Substituting the Jacobian at the interior rest

point yields

� �nX

i��

�i�i"xl � �l"xlfl��c�

� � �"sl�

nXi��

�i�i

�� �lgl � �l��"slflc� � �l�flc�"xl�� � "!� � � �

� �� �

Page 39: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

From �rst equation we obtain �l ��

flc�

nXi��

�i�i� Substituting this into the second

line yields

��

nXi��

�i�i

�"sl � "xl�� �

"!

flc�

� �lgl � �l��"slflc� � � �

Summing this equation over all l and using the normalizationP�xi�si� � � yields

after some further rearrangements

nXi��

�i

�� �

�ic�

nXl��

fl

�� � �

Since the terms in brackets are all positive� it is clear that there must be at least

one positive and one negative �� IfP

�i�i � � and �l�� is positive� we see from

previous equation that �l is positive as well� Applying this argument n times� we

see that all �l � � for l� which is a contradiction� IfP

�i�i � �� assume that

�l�� is negative for some l� The same argument as above shows that �i � �� i�e��Pi �i�i � �� Now �i � � implies �i � � for all i� The explicit expressions for �i

above prove that �i � � as well� Thus the Jacobian has rank �n and therefore it

is regular�

Proof of Theorem �� The equilibrium concentrations at the interior �xed point

can be expresses as

"si �"!� ki��fic�

� "xi �"!

c� ��� (���n�nX

l��

�i�l�i�l

with the abbreviations

�i�l ��

fi�l��

l��Ym��

gi�m� ( �

nYl��

gl� �i�l �

lYm��

�"!� ki�m� �

If there is an interior equilibrium then is ful�ls "! � k�� otherwise "s� would be

negative� Substituting the above expressions into the normalization conditionPi�"xj � "sj� � � yields

"!

�BB�����n� �

nPi��

nPl��

�i�l�i�l

�� (���n

�CCA�K���n� � c� � �

� �� �

Page 40: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

with K���n� �nX

l��

kl��fl

and ����n� �nX

l��

fl� We know from the discussion in sec�

tion ��� that the interior �xed point enters the simplex by means of a transcritical

bifurcation with PN � � Hence we can compute the value of "! at the bifurcation by

setting "sn � �� which implies "! � k� and we can explicitly compute the critical

value of c��

cN��N

� � k�

�����n� �

n��Xi��

n��Xl�i

�i��l

��K���n� � �

Since this is the only solution� no other �xed point can enter the interior of the

simplex� Moreover� we can verify by implicit di�erentiation of that "!�c� is

positive at c� � cN�N �

� and hence "! � k� if the total concentration is raised above

the critical value� Thus there is no interior �xed point for c� � cN�N �

� and exactly

one for c� � cN�N �

� � since the regularity of the Jacobian� proved in the previous

theorem� implies that the solution of the above system of equations is unique�

Proof of Theorem ��� �i� The equilibrium concentrations at the interior �xed

point are

"xi �r

"an � (���nnX

l��

�i�l"an�l�i�l � "si �

r � ki��"a

fi

with the abbreviations

�i�l ��

fi�l��

l��Ym��

gi�m� ( �

nYm��

gm� �i�l �

lYm��

�r � ki�m"a� �

The equilibrium concentration of the substrate "a is then implicitly determined by

the equation

F�a� r� def

��� a� a� �nX

i��

xi � si � � �

The function F has the explicit representation

F�a� r� def

��� a���K���n��� a� � r����n� �rPn

i��

Pnl�� a

n�l�i�l�i�lan � (���n

with the abbreviations K���n� �nX

i��

ki��fi

and ����n� �nX

i��

fi� The equilibrium

condition hence becomes F�"a�r�� r� � ��

� �� �

Page 41: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

�ii� Consider the equation an � (���n � �� It has a unique solution aP �r� in the

interval �� r�k� � which is strictly increasing in r and intersects the line a � a� at

a �nite value )r� We de�ne

S ���r� a�

���� r � �� � � a � mina�� r�k�� aP �r�

��

This set is obviously compact� The expression in item �i� imply that �"a� r� � Sfor an interior equilibrium� Since �F�a� is positive on S� there are at most twosolutions of F�a� r� � � on S��iii� There is exactly one �ux rate at which a solution of F � � intersects the linea � r�k�� namely

rN�N �

�a�k�

��K���n� � k�����n� � k�Pn��

i��

Pn��l�i �

�i���l

� �

The value of F is negative along the line a � r�k� for r � rN�N �

� For su�ciently

small � � � we observer F�aP �r�� �� r� � �� Thus there is at least one solution of

F � � in S for small �ux rates� Since the number of solutions must be odd butthere cannot be more than two we know that there is indeed a unique solution�

It is easy to check that the interior �xed point approaches PN � at the critial �ux

rate rN�N �

� Thus there is in fact a transcritical bifurcation�

�iv� In scenario �a� F is positive on the part of S to the right of the solution ofF � �� since F�r � � on S and hence F is monotonously increasing in r on S�Therefore� there is no interior �xed point for any �ux rate greater than rN�N �

�v� In scenario �b� there must be a saddle node bifurcation since rN�N �

is the

only �ux rate where a solution curve of F � � can intersect the boundary of S�From the monotonicity of F in r we conclude that there is exactly one saddle nodebifurcation�

Proof of Lemma � Suppose there is a �xed point PI with I �� N � Then thereis at least one index j such that �j � �� �� I implying "! � k� If j � � � I then

k�c�f"sj� "! � �� i�e�� c�f"sj � � which contradicts j � I� Repeating this argument

shows that i � I implies �i� �� �� I� and hence i � I implies also �i� �� �� I� The

same reasoning works for the CSTR model�

� �� �

Page 42: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

Proof of Lemma �� �i� Let us �rst consider constant organization� The equilib�

rium concentrations for �xed points on the boundary of the simplex are

"xi �k

jIj� � "si �g

jIj�

for all i � I� The Jacobian at �xed points on the boundary has jIj eigenvaluesc�f "s � � and hence they are all unstable�

�ii� The equilibrium concentrations of boundary �xed point PI in the CSTR model

are

"x �a�k � r

nk�� "s �

g�a�k � r�

n�� "a �

r

k

for all i � I� It is clear that these �xed points can exist only if r � a�k� The

Jacobian has the jIj�fold transversal eigenvalue f"s � �� The equilibrium PI is

therefore unstable �xed points are unstable�

Proof of Theorem ��� The coordinates of the interior �xed point are

"x �nk � c�f

n�n�� c�f�� "s �

g

n�� c�f�

The eigenvalues of the Jacobian can be obtained using the method outlined in

Appendix B� We �nd explicitly

����j �

��u�

pu� � �v � �w�cos�� � sin��

�for � � j � n� �

where � denotes the imaginary unit� � � ��j�n and

u �nk � c�f

nv �

c��f�g�nk � c�f�

n�n�� c�f��w �

c�fg�nk� c�f�

n�n�� c�f��

In addition we have the external eigenvalue�"! and the eigenvalue��n��c�f� � ��The eigenvalues corresponding to j � n�� �if n is even� are real and negative� For

j �� n�� the eigenvalues for j and n� j are complex conjugates� We are interested

in their real parts only� hence we only need to consider � � j � n��� The real

parts of the eigenvalues with the minus sign before the square root ���j � is always

negative� It is not hard to verify that ��� is has the largest real part of all the

eigenvalues ��j � Hence it determines the stability of the �xed point�

� �� �

Page 43: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

The eigenvalues ��j are all zero at c� � �� hence we can determine their sign for

small values of c� from the derivative�Re��jc�

�c���

�fg cos�

n��

Thus Re��� is positive for n � � and negative for n � � for small c�� For n � ��

examination of the second and third derivative show that the real part of ��� is

positive at small total concentrations if g � k and negative if g � k�

The real part of the eigenvalue changes sign if

c��f���� cos�� � c�fn

�k��� � cos��� g�sin� �� cos��

�� n�k� cos� � � �

For n � � there is no positive value of c� which satis�es the equation� The interior

�xed point is therefore always stable�

For n � � there are either two or Hopf bifurcations or none� They take place at

c� �g � �k �

pg� � ��gk � ��k��f

if the argument of the square root is positive� which requires that g � �����k� The

�xed point is stable for small total concentrations� becomes unstable in the �rst

Hopf bifurcation and stable again in the second�

For n � � the absolute coe�cient vanishes� i�e�� we encounter a Hopf bifurcation

at

c�� ���g � k�

f

if g � k� In this case the �xed point is a saddle below c�� and a sink above this

value� If g � k� the �xed point is stable at all values of c��

The quadratic coe�cient of is always positive% the absolute coe�cient is negative if

n � �� Therefore� if n � �� there is exactly one positive value c�� at which the real

part of ��� vanishes% since the �xed point is unstable at small total concentrations�

it becomes stable if c� exceeds this critical value�

Proof of Lemma �� The equilibrium concentrations at an interior equilibrium

are

"xi �r�r � k"a�

f�"a�� r�"si �

r � k"a

f

� �� �

Page 44: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

where "a is a solution of the equation

"a���nk � f� � "a��fa� � nr� � fr � fa�r � � �

Such a solution is physically meaningful if a � a� and r�� � "a � r�k� The lemma

follows easily from these conditions�

Proof of Theorem ��� As for the constant organization case we can analyze the

Jacobian matrix using the method described in appendix B� We �nd the external

eigenvalue �r and two eigenvalues which do not change sign with the �ux rate�We do not give them here since their explicit expressions are very complicated� It

can be shown� however� that one of them is always negative� while the other one

is positive in one and negative in the other �xed point whenever there are two

interior rest points P���N and P

���N �

The remaining eigenvalues are of the form

����j �

��u�

pu� � �v � w�cos�� i sin��

�for � � j � n� �

with the abbreviations

u � f "x� r � v � f "x�r � k"a� � w � fg"x"a

and � � ���j�n� There can be Hopf bifurcations for all ��j with j �� n��� but

since ��� has the greatest real part� only Hopf bifurcations at this eigenvalue change

the stability of the �xed point�

For n � �� the real part of ��� changes sign at

r� �a�f

��� �fg � ��fk � ��gk � ��k

� � ��f � �k�pg� � ��gk � k�

�f� � fg � �fk � �gk � �k�

There are either two Hopf bifurcations or none �the latter case occurs if r� is

negative or complex��

For n � � a Hopf bifurcation takes place if g � k� The critical value of the �ux is

r� �a�f�g � k�

��f � ��g � k��

The �ux rate at which the real part of ��� changes sign can be determined for

arbitrary n� but the general formulae are too complicated to be reproduced here�

� �� �

Page 45: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

It can be shown� however� that for n � � there is exactly one �ux rate at which

��j changes is sign�

If f � f c there is another technical problem� Since the equation for the eigenvalues

above has the same form for both interior �xed points it does not readily tell us

at which one of them a Hopf bifurcation occurs� In order to decide this question

we need to explicitly compute the equilibrium concentration of the substrate at

the Hopf bifurcation�

At r � �� "a � � and at r � a�k� "a � a�� all eigenvalues ��j are zero� The sign of

the eigenvalue near these critical �ux rates can be determined from the sign of the

derivative Re���j ��r� We �nd

�Re���j �

r

�r���a��

�g

��cos�� �� �

This is negative for all j and hence ��j is negative for su�ciently small �ux rates�

Near the transcritical bifurcation we �nd�Re���j �

r

�r�a�k�a�a�

�fg

fg � nk�cos� �

The denominator is positive if f � f c� that is� if there are two interior rest points�

In this case the sign of the real parts of the eigenvalues equals the sign of cos ��j�n

for �ur rates r that are not much larger than a�k� If f � f c the denominator is

negative and the sign of the real part of the eigenvalue equals the sign of cos ��j�n

for �ux rates not much smaller than a�k�

Appendix B� Eigenvalues of the Jacobian

In this section we brie�y discuss the properties of the Jacobian matrix at an in�

terior equilibrium for the symmetric models� Not much can be said about more

general cases� unfortunately� The computations have been assisted by the sym�

bolic mathematics packages Reduce and Mathematica� Since most of the explicit

expressions are very complicated we present only a brief outline�

� �� �

Page 46: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

The Jacobian for the system under constant organization has the entries

Uijdef

��� �xixj

� Vijdef

��� �xisj

� Wijdef

��� �sixj

� Xijdef

��� �sisj

The matrices U� V�W and X have size n� n� In the CSTR we have additional

terms of the form

�pidef

��� �xia

� �qidef

��� �sia

� �videf

��� �a

xj� �wi

def

��� �a

sj� z def

��� �a

a�

The Jacobian can thus be represented in the form

Jc�o� �

U V

W X

and Jcstr �

�� U V �pW X �q�v� �w� z

�A �

respectively� The crucial observation is that the quadratic n�n blocks are circulantfor the interior �xed point of both the symmetric cooperation and the symmetric

comptetition model� The vectors �p� �q� �v and �w are multiples of the vector � �

��� � � � � �� in this case� Let I be the identity matrix and let J denote the matrix

with all entries �� The matrix C has the entries �i�j�� where the indices are taken

modulo n� The zero�matrix is denoted by O� Clearly� the matrices I� J� O� C and

its transpose C� are circulant� With this notation we �nd explicitly�

competitive model under constant organization

J �

��"xJ c�f "xI��"sJ� g��"sc�f�I ��c�f "x� "!�I

cooperative model under constant organization

J �

��"xJ �cofC��"sJ� gI� c�f"sC

� ��c�f "x� "!�I

competitive model in the CSTR

J �

�� O f "xI k"x��g"a� f"s�I ��f "x� r�I g"x���"x�� �� ��r � n�"x�

�A

cooperative model in the CSTR

� �� �

Page 47: Autocatalytic Networks with Intermediates I: Irreversible ... · Intermediates I: Irreversible Reactions Robert Hecht Robert Happel ... elemen trary steps of simple replicator dynamics

Hecht et al�� Autocatalytic Networks with Intermediates

J �

�� O f "xC k"x�g"aI� �k"a� r�C� ��f "x� r�I g"x�

��"x�� �� �r � n�"x�

�A

The vectors

�y�j� � ��� �j� ��j� �j� � � � � ��n���j� with � � exp�����n� and � � j � n

are eigenvectors of all circulant n � n matrices� They can be used to explicitly

construct eigenvectors of the above Jacobian� In particular� the ansatz ��j� �

��y�j�� �j�y�j�� leads to a quadraric equation for the parameter �j in the constant

organization case� In general we obtain two solutions for �j belonging to two

di�erent eigenvalues of J� The details of this procedure for matrices consisting of

m�m circulant blocks of size n� n has been described in detail in �� � Explicit

expressions for the eigenvalues are used in some of the proofs in appendix A�

The situation is similar for the CSTR case� where we have an additional row and

column� An appropriate ansatz for the eigenvectors of J is ��j� � ��y�j�� �j�y�j�� �j��

The procedure is simpli�ed considerably by the observation that �y��� � � and that

the vectors �y�j� are orthogonal� A short computation then shows that �j � � for all

j �� �� We therefore obtain a quadratic equation for �j which completely analogousto the constant organization case� For j � � we obtain a non�linear system of

equations in the variables �� and ��� Eliminating �� we obtain a cubic equation for

��� One root is the external eigenvalue �r� The remaining quadratic equation canbe solved explicitly� In general we obtain very complicated expressions� however�

which we do not reproduce in this contribution� They can be found in the doctoral

thesis �� �

� �� �