Autocatalytic Replication in a CSTR and Constant Organization · CSTR and Constant Organization Robert Happel Peter F. Stadler SFI WORKING PAPER: 1995-07-062 ... a mathematical theory

Autocatalytic Replication in aCSTR and ConstantOrganizationRobert HappelPeter F. Stadler

SFI WORKING PAPER: 1995-07-062

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SANTA FE INSTITUTE

Autocatalytic Networks with Translation

Robert Happela� Robert Hechta� and Peter F� Stadlera�b��

aTheoretische Biochemie� Institut f�ur Theoretische Chemie

Universit�at Wien� Vienna� Austria

bSanta Fe Institute� Santa Fe� NM

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

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

E�Mail� studla�tbi�univie�ac�at

Happel et al�� Autocatalytic Networks with Translation

Abstract

We consider the kinetics of an autocatalytic reaction network in which replication and catalytic action is separatedby

a translation step� We �nd that the behavior of such a system is closely related to second order replicator equations�

which describe the kinetic of autocatalytic reaction networks in which the replicators act also as catalysts� In fact�

the qualitative dynamics seems to be described almost entirely be the second order reaction rates of the replication

step� For two species we recover the qualitative dynamics of the replicator equations� Larger networks show some

deviations� however� A hypercyclic system consisting of three interacting species can converge towards a stable

limit cycle in contrast to the replicator equation case� A singular perturbation analysis shows that the replication

translation system reduces to a second order replicator equation if translation is fast� The in�uence of mutations

on replication translation networks is also very similar to the behavior of selection�mutation equations�

�� Introduction

A variety of self�replicating chemical systems have been constructed and investigated experimen�

tally in the past �� years since Spiegelman�s �� in vitro serial transfer experiments on Q�� The

dynamics of this model�system� which contains a protein enzyme as part of the environment��

was subsequently studied in great details by Manfred Eigen and his collaborators �� Cat�

alytic activity of RNA for a variety of reactions has been discovered �� including a

limited capacity for replication and self�replication �� Arti�cial molecules that self�replicate

in a protein�free environment have been synthesized� for instance� by Orgel �� Rebek ��

�� and von Kiedrowski �� All these �ndings add to the credibility of an RNA world� as one

of the crucial stages in prebiotic evolution ��

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

has been developed� based on a series of pioneering papers by M� Eigen and P� Schuster ��

�� Ignoring the molecular details one considers the dynamics of replicators as a auto�catalytic

chemical� reactions of the form

�A� � I � �I � �W ��

where I is the replicator� �A� denotes the energy�rich building material for replicators and �W �

indicates that the formation of the replicator will in general produce low�energy waste��

One of the central questions in any theory of molecular evolution concerns the behavior of a collec�

tion of competing �or otherwise interacting� species of replicators I�� In� While the replication

of polynucleotides is in reality an overwhelmingly complicated process involving a huge number

� � �


of intermediates it nevertheless has turned out that it is sensible to focus on the over�all replica�

tion reaction� In other words� one can treat the replicators as atomic� entities� and encapsulate

their chemical details into a small set of reaction rate constants �� The system of interacting

replicators is thus completely described by the concentrations �or relative frequencies� xk of the

di�erent species Ik� In its most condensed form the logics of the RNA world is thus captured in

the autocatalytic reaction network

Ik � Il �� Ik � Il k� l � �� n�

The dynamical system associated with the above reaction scheme is now termed replicator equation

��

�xk � xk

�� nX

j��

akjxj �Xi�j

aijxixj

�A � �R�

Originally developed as a model of prebiotic evolution� replicator equations have been encountered

since then in many di�erent �elds� populations genetics� mathematical ecology �where they occur

disguised as Lotka�Volterra equations �� economics� or laser physics� Their properties have been

the subject of hundreds of research papers by many research groups� most prominently among them

J� Hofbauer� P� Schuster� and K� Sigmund in Vienna� The results of the �rst decade of investigation

are compiled in the book ��

Self�replication on molecular level was clearly the crucial invention at the origin of life� Almost

equally important was the invention of translation and the genetic code� In this contribution we will

not be concerned with the question why translation was invented� or how a RNA replicator that has

invented translation would conquer the RNA world �� We have a much less ambitious agenda

here� We consider a reaction system that contains a number of di�erent replicator species which all

produce a gene product that carries the enzymatic activity� Our main question is the following� Is

the picture of the selection dynamics that has emerged from the studies of autocatalytic reaction

networks still valid once the catalytic activity has shifted to translation products�

� � �


��The Model

We consider a system of n species I�� In of replicators or genotypes and their translation

products T�� Tn� For sake of de�niteness� we may consider the genotypes as nucleic acids and

the translation products as proteins in same later stage of prebiotic evolution �� The replication

processes involve translation products Tj of the genotype Ij as catalysts�

�A� � Ik � Tja�kj�� Ik � Tj � �W �

for all combinations � � k� j � n� The rates of these reactions are a�kj��A�� Ik� �Tj�� i�e� we assume

mass action kinetics� Since there are no convincing models for the kinetics of translation in a

prebiotic setting we make the most simple choice�

�B� � Ikw�k

��Tk � Ik

with rate w�k��B��Ik�� We assume that the genotypes Ik and the translation products are di�erent

types of biopolymers� thus the concentrations of the monomers ��A�� and ��B�� are independent

from each other� Consequently we will assume that the two types of polymers are subject to

di�erent degradation or removal reactions� In the mathematical model the latter are described by

out�ows �Tk� T and �IK � � respectively�

Thus the dynamical system considered here reads in its most general form

d

dt�Ik� � �Ik�

��

nXj��

a�kj��A�� Tj��

��

d

dt�Tk� � w�

k��B�� Ik�� Tk� T

Note that we are still missing equations governing the concentrations of the monomers ��A�� and

��B�� and that we have not yet speci�ed the functions and T which describe the removal

processes� It is convenient to consider these as boundary conditions� under which the replication�

translation process takes place� Three types of boundary conditions have been discussed in the

literature�

�� Constant Organization assumes �i� that the monomer concentrations are bu�ered and thus

constant in time� and �ii� that the total concentration of the di�erent polymer types is kept

� � �


Computer

(A)(B)(solvent)

measurements

Figure �� The hypothetical Evolution Reactor is a kind of dialysis reactor with walls impermeable to polynu�

cleotides and polypeptides� The in��ow of monomers keeps the reaction mixture away from equilibrium�

It is adjusted such that the the concentrations of the monomers occur in excess in the reactor� i�e��

the monomer concentrations do not change signi�cantly over time� The out��ow of polynucleotides

and poly�peptides through two di�erent diaphragms is controlled such that total concentration of both

types of polymers is kept at constant levels� While this type of evolution reactor will not be easy to

realize in practice it establishes boundary conditions for replication�translation systems that lead to

mathematically tractable dynamical systems�

constant in time� These conditions can in principle be enforced in a sophisticated evolution

reactor as the one shown in �gure �� It will be convenient to use the e�ective rate constants

akjdef

�� A��a�kj and wkdef

�� B��w�k �

A further simpli�cation is achieved by switching to relative coordinates

xkdef

�� Ik�X

j

�Ij� and tkdef

�� Tk�X

j

�Tj� �

As simple calculation then shows that

�Xi�j

xiaijtj and T �Xi

wixi �

In matrix notation we obtain thus

�xk � xk ��At�k � hx�Ati� �tk � wkxk � tkhw� xi �CO�

� �


The state space of this model is the direct sum of the two �n � ��dimensional simplices

corresponding to the coordinates xk� k � �� n� and tk� k � �� n� respectively�

�� The Continuously Stirred Flow Reactor �CSTR� is the most convenient experimental setting�

A constant �ow of rate r through the system carries the monomers with concentrations a�

and b� and removes monomers and polymers at the same �ow rate r proportional to their

concentration� Using conservation of mass one immediately �nds the dynamical equations

for the concentrations of the monomers a def

�� A�� and b def

�� B�� In order to simplify the

notation we use ykdef

�� Ik� and uk � �Tk��

�yk � yk �a�A�u�j � r�

�uk � w�kbyk � ruk

�a � �ahy�A�ui� r�a� � a�

�b � �bhw�� yi � r�b� � b�

�CSTR�

The state of the system is described by a vector �y� u� a� b� � IRn� � IR

n� � IR� � IR��

�� We assume that there is no �ux of material into or out of the system� The system is kept away

from thermodynamic equilibrium by means of an energy consuming regeneration reaction that

produces active monomers from the degradation products of the polymers� While not very

realistic for an experimental approach this type of boundary condition is very useful for the

study of pattern formation processes as it leads to meaningful reaction� di�usion models

�� In spatially homogeneous models we obtain equations similar to the CSTR

discussed above� The rate constants of the degradation reactions are dk and dTk � respectively�

Conservation of mass ensures that ��A�� P

k�Ik� � a� and ��B�� P

k�Tk� � b��

�yk � yk �a�A�u�j � dk�

�uk � w�kbyk � dTk

�a � �ahy�A�ui� hd� yi

�b � �bhw�� yi � hdT � ui

�REG�

The state space is the direct sum Sn�� Sn�� of two n�dimensional simplices�

Since all three dynamical systems describe the same chemical reaction system� although under quite

di�erent boundary conditions it is not surprising that their dynamics is quite similar� Constant

organization is by far the most tractable case� We will therefore give a complete derivation of

� � �


our results only for this case� Results for the CSTR and the regeneration system are often very

similar to the CO case� The details can be found in �� It has been observed quite often that the

dynamics of replication systems in a CSTR or in a model with regeneration reaction�s� become

very similar to the constant organization case when the �ow rate r �or the reaction rates for the

regeneration steps� become small� A singular perturbation treatment of this e�ect can be found

in �� For a partial result see ��

�� Fixed Points

Despite the fairly complicated form of our reaction scheme it is not di!cult to compute the coor�

dinates of all equilibria� Let us begin with interior equilibria� that is� with rest points at which all

species occur with non�zero concentration� We shall use the notation � for the vector with entries

��

Theorem �� Consider the reaction translation model under constant organization� Then there is

a unique interior equilibrium �"x� "t� � int �Sn � Sn� if and only if A�� is either strictly positive or

strictly negative�

Proof� Suppose A is invertible and �"x� "t� is a �xed point of �CO�� An explicit computation shows

"xk ��A��k

wk

nXi��

�

wi�A��i

and "tk ��A��kh�� A��i

�

Thus "t � intSn if and only if �A��k has the same sign for all k� Then "xk � � as well since the

coe!cients wk are all strictly positive by assumption�

Remark� If there is an isolated interior equilibrium� then A is invertible and �A��k has the

same sign for all k� Furthermore there are at most two isolated interior rest points for CSTR�

The expression for the x�coordinates of the interior rest point above strongly suggests to introduce

the matrix

B def

�� Adiag�w��

With this de�nition we may write

"x ��

h�� B��iB�� and "t �

�

h�� A��iA��

� �


for the location of the interior equilibrium�

On the boundary of the state space at least one coordinate is zero� It is useful to observe that

equilibria on the boundary have a particular form�

Lemma �� Suppose all translation rates are non�zero and let � � �"x� "t� be a rest point� Then

"xk � � if and only if "tk � ��

Proof� Suppose "xk � �� Then �tk � �tkhw� xi� where the scalar product hw� xi � � by assumption�

Thus "tk � �� Now suppose "tk � �� This implies � � wk"xk and thus "xk must vanish�

If some of the wk are zero� parts of be boundary consist entirely of �xed points� We will not

consider these degenerate cases any further� The non�zero coordinates of � can be obtained by

restricting the dynamical system to the variables that do not vanish in �� i�e�� to a smaller system

of the type �RT�� The theorem above can therefore be applied also to the non�zero part of a rest

point � on the boundary of the state space� All isolated rest points of �RT� are thus obtained as

the interior rest points of �RT� restricted to a subset K � f�� ng of the n replicating species�

The stability analysis of the rest points will turn out to be very complicated in general� It is

fairly easy� however� to determine the stability of a boundary equilibrium � against introduction

of species which are not present in �� The corresponding directions are called transversal �� To

this end it will be convenient to temporarily rearrange the order of the coordinates such that

� � �x�� t�#x�� t�# � � � #xn� tn�

and "xi � "ti � � for � � i � m� The entries of the Jacobian matrix in the �rst �m rows and

columns are readily computed�

� �xk�x�

�� kl��A"t�k � h"x�A"ti� � �def

�� Lk��kl

� �xk�t�

��

� �tk�x�

�� klwk

� �tk�t�

�� hw� "xi

Consequently the Jacobian �RT�� is of the form

� � �


�BBBBBBBBBBB�

�L�� w� hw� "xi

��

� � � � � ��

�

�L�� w� hw� "xi

��

� ��

��

��

� � � � �

�Lm�� wm hw� "xi

��

X Y

�CCCCCCCCCCCA

where the ��n�m��m block X is irrelevant for the stability of � and Y is the ��n�m��n�m�

Jacobian matrix of �RT� restricted to the species that do not vanish in the equilibrium �� The

eigenvalues corresponding to the transversal direction k are now easily computed from the ��

blocks� We �nd explicitly

��k � Lk�� and �

��k � �hw� "xi � ��

Assuming as usual hw� "xi �� we have "t � �hw��xidiag�w� "x and thus

��k �

�

hw� "xi��Adiag �w� "x�k � h"x�Adiag �w� "xi� �

�

hw� "xi��B"x�k � h"x�B"xi� �

Following �� we will say that a rest point � of �RT� is saturated if all eigenvalues belonging to

the transversal directions are non�positive� The above considerations can then be summarized as

Theorem �� Let � � �"x� "t� be a boundary equilibrium of a replication translation model for which

A is invertible and wk � � for all k� Then � is a saturated equilibrium if and only if "x is saturated

equilibrium of the second order replicator equation with interaction matrix B � Adiag�w�� i�e�� if

and only if

$�k��def

�� B"x�k � h"x�B"xi � � for all k with "xk � ��

The matrix B � Adiag�w� occurs both in the explicit expression for the x�coordinates of an

interior rest point and in the expression for the transversal eigenvalues of a boundary equilibrium�

In both cases our result match the situation in a second order replicator equation with B as

interaction matrix� We will see in the following sections that this relation between �RT� and

replicator equations is even deeper�

In �� it has been proposed to represent an autocatalytic network by a directed graph with colored

edges� In complete analogy we introduce the same notation here� The vertices of the graph %�RT �

� � �


b > bjjij

Commensalism Parasitism

b bjjijb b jjij= <

MutualismSymbiosis

ji i i

i i

i

j j

j

j

Amensalism

Competition

Neutralism

j

>b b

b b

b b

=

<

ji ii

ji ii

ji ii

Predator-Prey

Figure �� The di�erent types of interaction� full arrows indicate bij � bii � �� empty arrows bij � bjj ��

associated with the reaction�translation model �RT� are the replicating species I�� In� There is

a �full� arrow from j to i if bij � bii � �aij � ajj�wj � �� and there is a �empty� arrow from j to i

if bij � bjj �� As an example� look for the two�species model as follows� The six di�erent graphs

of this type on n � � vertices match the usual classi�cation of ecological interactions between two

species� see �gure �� In the following section we show that they also correspond in a very natural

way to the classi�cation of the dynamical behavior of the two�species replication�translation system�

see �gure ��

��Two Species

The special case n � � allows for a complete analysis of the dynamics under constant organization�

We consider here the general two�species system with general selection matrix A with entries

aij � � and and translation constants �w�� w�� It will be convenient to use the following

abbreviations�

c�def

�� a�� a�� c�def

�� a�� a��

The results of the previous section imply immediately that an interior equilibrium exists if and

only if c�� c� � � or c�� c� �� The Jacobian of the interior rest point can be readily diagonalized

� � �


with the help of Mathematica� One �nds the two external eigenvalues

�ext� �D

��c� � c��and �ext� � �

w�w��c� � c��

c�w� � c�w�

which do not in�uence the dynamical behavior on S� � S�� Both are negative whenever there is

an interior rest point� The dynamical stability of this �xed point is determined by the remaining

two eigenvalues

�� c� � c��w�w�

p�c� � c��w�

�w�� c�c�w�w��c�w� � c�w��

��c�w� � c�w��

R

RT :

:

(x ,t ) (x ,t )

(x )

(x ,t )(x ,t )

(x )1 2

1 1 1

2 1 2 2

2

Figure �� Comparison of the phase portraits of �RT� and �R�� Black circles indicate sinks� gray ones saddle�points�

and white ones are sources� The corresponding graph %�RT � are shown below�

This is of the form �p�� where � is always negative and sgn� � sgnc�� provided c� and c�

have the same sign� Assume that c�� c� � �� Then the square root is either complex or if it is real

then it is smaller than �� Consequently �� and the interior rest point is a sink� If c�� c� � �

then � � and the square root is real and larger than �� hence � � and � � �� and the interior

equilibrium is a saddle point�

Finally� let us brie�y consider the ��species equilibria� Their stability is determined by the eigen�

values�� # ��

a��c�c� � c�

� �� c�c�

c� � c��

�� # �� a��c�c� � c�

� �� c�c�

c� � c��

Summarizing our calculations we have

� ��


c�� c� � Both corners are sinks and the interior equilibrium is a saddle point� Its unstable

manifold connects to both sinks�

c� � �� c� � Corner � is a sink and corner � a saddle point� The unstable manifold of corner �

connects to the sink� There is no interior rest point�

c� �� c� � � Corner � is a sink and corner � a saddle point� The unstable manifold of corner �

connects to the sink� There is no interior rest point�

c�� c� � � Both corners are saddle points and the interior equilibrium is a sink� The unstable

manifolds of both corner saddle points connect to the interior sink

The phase portraits corresponding to these four cases are shown in Fig� �� They compare directly

to the phase portraits of the second order replicator equation with interaction matrix A�

��Barycentric Transformation

Before we proceed with the linear stability analysis of interior equilibria in larger systems� we

brie�y discuss a transformation that will turn out to be a crucial tool for most of our results� The

following lemma is well known� see e�g� �� sect� ��

Lemma �� Let D � diag�d�� dn� be a diagonal matrix with di � �� Then there is a di�eo�

morphism B � Sn � Sn mapping the phase portrait of the second order replicator equation with

interaction matrix A to the phase portrait of the second order replicator equation with interaction

matrix AD�

This result can used to simplify the algebra for the stability analysis of an interior rest point�

Suppose there is an interior rest point "x� Setting didef

�� "xi sends "x results in B�"x� ��n�� i�e��

the interior rest point is mapped to the barycenter of the simplex� Therefore B is usually called

barycentric transformation� The fact that the coordinates of the interior rest point are now of a

very simple form simpli�es the algebraic manipulations in many cases� Fortunately� a similar result

holds for our replication�translations system �RT��

� ��


Theorem �� Let aij be an arbitrary matrix� �x� t� � Sn � Sn and let �c� d� � int �Sn � Sn�� Then

B � Sn � Sn � Sn � Sn� �x� t� � �u� v� de�ned by

ui �cixiPj cjxj

and vi �ditiPj djtj

is a di�eomorphism mapping the orbits of

�xk � xk

��X

i

akiti �Xi�j

xiaijtj

�A

�tk � wkxk � tkXi

wixi

onto the orbits of

�uk � uk

��X

i

bkivi �Xi

uiXj

bijvj

�A

�vk �

��kuk � vk

Xj

�juj

�A ��u� v��

where the coecients of the latter di�erential equations are bij � aij dj� �i � diwi ci� and

��u� v� �

Xl

vldl

��Xl

ulcl

� ��

Proof� The inverse map B�� is given by

xi �uici

�Pj c

��j uj

and ti �vidi

�Pj d

��j vj

�

Di�erentiating ui and vi yields

�ui ��P

j cjxj

�� xiciX

j

cjxj � cixiXj

cj �xj

��

� ui

��X

j

aijtj �Xj

xjXl

ajltl

�A �

� ui

��X

j

aijdj

vj �Xj

ujXl

ajldlvl

�A �P

j d��j vj

�vi �di �tiPj djtj

� viXj

dj �tjPl dltl

�diwixiPj djtj

� viXj

djwjxjPl dltl

�

��diwi

ciui � vi

Xj

djwj

cjuj

�APj d

��j vjP

j c��j uj

By means of a change of velocity and setting bij � aij dj and �i � diwi ci we obtain �nally the

proposition�

� ��


Corollary �� Let "� � �"x� "t� be an equilibrium of �CO� and let

ui �xi

"xiP

j "x��j xj

and vi �ti

"tiP

j"t��j tj

�

Then the �xed point of the dynamical system obtained from �CO� by a barycentric transformation

with parameters ui� vi is "B�"�� n��

Proof� From theorem � one �nds immediately

"ui �"xi

"xiP

j "x��j "xj

��

nand "vi �

"ti"tiP

j"t��j"tj��

n�

��Competitive Systems

Schl�ogl �� investigated two model systems in which the substance Xi is formed from a substrate A

via �rst order and second order autocatalysis� respectively� A�Xi � �Xi ��rst order autocatalysis�

and A � �Xi � �Xi �second order autocatalysis�� In the �fully� competitive case the translation

products catalyze only the replication of their own gene� i�e�� the interaction matrix is diagonal

A � diag�k�� k�� kn�� Therefore the di�erential equations for the competitive model simplify to

�xk � xk

��kktk �X

j

kjxjtj

�� tk � wkxk � tk

Xj

wjxj

The phase portraits of this dynamical system as equivalent with the phase portraits of

�uk � ui

��vi �X

j

ujvj

�� vk � �uk � vk��u� v�

as a consequence of the barycentric transformation�

The Jacobian matrix of this vector �eld is readily computed�

�

�uj�ui � �uivj

�

�vj�ui � �uiuj � ui�ij

�

�uj�vi � �ui � vi�

��

�uj� �ij��u� v�

�

�vj�vi � �ui � vi�

��

�vj� �ij��u� v�

� ��


At the �xed point P � �n�� we �nd that ��P� � w� Since ui � vi at this point� the values of

�� ui and �� vi are irrelevant� The Jacobian is of the form

�f�P� �

�A BC D

�

where each of the four quadratic matrices A� B� C� and D is circulant� We �nd explicitly�

A � ��

n�J � B � wI � C � �

�

n�J �

�

nI � and D � �wI �

where J is the matrix with all entries one and I is the identity matrix� Matrices of this type can

be analyzed using the following interesting result�

Theorem �� Let M be a mn �mn matrix which has m� circulant blocks Mij of size n� n� The

vectors ��k� with entries

��k�j � exp��

j

k� � for k � �� n� � and j � � � � �n

are well known to be eigenvectors of any circulant n � n matrix� Set Mij��k� � �

�k�ij ��k�� let R�k�

be the m �m matrix with entries ��k�ij � and denote the eigenvalues of R�k� by &�k��

Then &�k�� is an eigenvalue of M � In particular� if all matrices R�k� are diagonalizable� then we

obtain all eigenvalues of M as eigenvalues of the matrices R�k��

Proof� We rely on the fact that the ��k� form a basis of eigenvectors for all circulant matrices�

Let z � IRm and suppose � � z � ��k� is an eigenvector of M � Then

M� �

�BB�

M��z��k� M��z��

�k� � � � M�mzm��k�

M��z��k� M��z��

�k� � � � M�mzm��k�

��

��Mm�z��

�k� Mm�z��k� � � � Mmmzm�

�k�

�CCA � �R�k�z�� k� � &�k��

Since ��k� is non�zero� this equation is equivalent to the eigenvalue equation for the matrix R�k��

Thus � is an eigenvector of M if z is an eigenvector of R�k�� and all eigenvalues of R�k� are also

eigenvalues of M �

Now suppose that all matrices R�k�� k � m are diagonalizable� Denote by z�k�� n� an

orthonormal eigensystem of R�k� and let ��k�� def

�� z�k�� k� be the corresponding eigenvectors

of M � We �nd that

hz�k�� k�� z�k�� k��i � hz�k��z�k

��i h��k��k��i � �� k k� �

� � �


i�e�� all eigenvectors corresponding to di�erent k are orthogonal� even if the corresponding eigenval�

ues should coincide by chance� Thus the ��k�� form a complete orthonormal basis of eigenvectors

for M �

In the terminology of theorem we �nd

R��

��

n �w �w

�and R�j� �

��

nw �w

�

for the competitive model� R�� belongs to the external directions since �� The correspond�

ing eigenvalues are

&��

nand &�� w�

The remaining eigenvalues &�j�� for � � j n and � � �� are now easily computed using

theorem �

&�� w

��

�

pw� � w n

independent of j� Both &�� and &�� are n � ��fold degenerate� and it is easy to check that

&�� &�� for all n � � and w � �� Thus � is a saddle point with n�� unstable eigenvalues�

The result immediately carries over to all equilibria on the boundary of the state space in the

following form� Let � be a rest point on the boundary and suppose there are m non�vanishing

species at �� Then � has m � � positive and m � � negative eigenvalues which all belong to the

directions spanned by the non�vanishing species� In particular� if � is stable� then m � �� i�e��

only the single�species equilibria can be stable� That they are in fact stable is determined by the

transversal eigenvalues �k�� w ��

��Cooperation

The hypercycle �� may serve as the paradigm of a cooperative system� Each translation product

catalyzes the replication of one other species in a circular arrangement� see �gure # consequently�

the interaction matrix is circulant� and the corresponding system of di�erential equations reads�

�xk � xk

��kktk��

Xj

kjxjtj��

��

�tk � wkxk � tkXj

wjxj �

� ��


Again a barycentric transformation yields

�uk � ui

��vi �X

j

ujvj��

��

�vk � �uk � vk��u� v� �

The external eigenvalues of the interior equilibrium �� are

&��

nand &�� w �

For the remaining eigenvalues we �nd

&��j � �

w

��

�

rw� �

w

ne��j��n �

and their real parts can be readily computed�

�&j� � �w �

rw

�

sw �

ncos��

rw� �

�w

ncos� �

�

n��

It is not complicated now to derive the critical value of w at which a Hopf�Bifurcations occurs�

��

w

�crit

� ��

n

sin��n� ��j

n

�

cos

��n� ��j

n

� �

Tn 1

I2

T1I

T2

I

T3

3

I4

In

a)

1

2

3

4

n

b)

Figure �� The hypercycle� a� chemical reaction scheme� b� graph representation�

� � �


Since w � �� only values of j in the interval n j �n can ful�l this condition� Consequently�

there are no Hopf bifurcations in the two�species model� For larger systems we �nd

n � �� j � � wcrit � � ��

n � � j � � wcrit � �� Since we consider only non�zero values of the translation rate� there is in

fact no stable �xed point� The corresponding hypercyle equation� however� exhibits

a marginally stable �xed point ��

n � � There are no stable interior �xed points� Computational studies indicate that there is

a �globally� stable limit cycle� as in the case of the second order replicator equations

��

All equilibria on the boundary are non�saturated and degenerate� in complete analogy to the

situation in the elementary hypercycle ��

�Reduction to Replicator Equations

A good introduction to singular perturbation theory are the books �� The results outlined

in the following few paragraphs are well known� see e�g�� We compile them here in

order to make this contribution self�consistent� Consider the singularly perturbed problem

�x � f�x� y� a� ��

� �y � g�x� y� a� ��SPP �

where x � X and y � Y � a � A � IRp is the admissible space of parameters� and � � I � R�

Furthermore let K � X � Y be compact such that int K is simply connected� We are interested

in the dynamics in the compact set K�

Suppose g has the following properties�

�i� There is a unique function � � X �A� Y such that g�x� ��x� a�� a� ��

�ii� The Jacobian J�x� a� ��g

�y�x� ��x� a�� a� �� is uniformly hyperbolic on A � X� i�e�� there is

a positive constant � � � such that absolute value of all eigenvalues of J�x� a� is bounded

below by � for all x � X and all a � A�

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

� ��


Theorem �� Under the above hypotheses there are open sets A� � A� I� � I such that for all

�a� �� A��I � there is a unique integral manifoldMa�� fy � ��x� a� ��jx � Xg with the following

properties

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

�ii� � satis�es uniformly on X � A�

lim��

��x� a� �� x� a� and lim��

��

�x�x� a� ��

��

�x�x� a� �

If J�x� a� is stable on X�A then the long�time behavior of a trajectory passing through a point x�

in a suitably small neighborhood ot the integral manifoldMa�� is determined by the dynamics on

this manifold� i�e�� by the di�erential equation �x � f�x� ��x� a� �� a� �� Under these circumstances

we introduce the notation

F �x� a� def

�� f�x� ��x� a�� a� ��

'�x� a� �� def

�� f�x� ��x� a� �� a� �� f�x� ��x� a�� a� ��

The di�erential equation �x � F �x� a� is known as the degenerate system� Property �ii� of � means

that there is a continuous function �� with �� such that k��x� a� �� x� a�k� ��

Here k � k� denotes the C� norm� see� e�g� �� p� �� If f is continuously di�erentiable with

uniformly bounded derivatives on X � Y there is a continuous function �� with �� such

that k'�x� a� ��k� �� In other words� the dynamics of trajectories near the integral manifold

Ma�� is described by the di�erential equation

�x � F �x� a� � '�x� a� ��

where '�x� a� �� is a regular perturbation of the degenerate system �x � F �x� a�� In such a case we

will say the the singularly perturbed problem �SPP� reduces to the degenerate problem�

These facts from singular perturbation theory make precise in what sense our replicaton�translation

system is related to replicator equations� We set wk � �k � and consider the limit wk � �� i�e�

�� with constant �k and �nd the following very general correspondence�

Theorem � The replication translation model �CO� reduces in the singular limit wi �� to the

second order replicator equation with interaction matrix bijdef

�� aij�j�

� ��


0.1 0.2 0.3 0.40.0

0.2

0.4

0.6

0.80.0 0.1 0.2 0.3 0.4 0.5

0.0

0.2

0.4

0.6

0.8

0.1 0.2 0.3 0.40.00

0.20

0.40

0.60

0.800.1 0.2 0.3 0.4

0.00

0.20

0.40

0.60

0.80

a) b)

c) d)

Figure �� The close relations between �R� and �RT� are perhaps best exempli�ed by strange attractors�

Chaotic attractors have been found in three�species Lotka�Volterra equations �� Using

Hofbauer�s transformation �� they can be translated to four�species second order replicator equations�

A two�parameter family of strange attractors with interaction matrix

A��

�B�

� ��

�CA

has been studied in detail � �� The numerical examples shown here correspond to � � ��

and wk � w for all k�

The graphs show a two�dimensional projection of the x�coordinates� a� slow translation w � �� b�

w � �� c� very fast translationw � �� and d� second order replicatior equation�

Proof� Let wk � �k � with �k� � � �� With the notation we obtain the singular perturbation

problem

�xk � xk

��X

i

akiti �Xi

xiXj

aijtj

�A

� �tk � �kxk � tkXi

�ixi

The fast time scale yields tk�x� ��kxkPi �ixi

in the limit � � �� It is easy to see that this solution

is stable for all x�

� �tk�tj

��t�x�

� ��

��kjXi

�ixi�

� ��


and hence the singular perturbation problem reduces to

�xk ��P

j �jxjxk

��X

i

aki�ixi �Xi�j

aij�jxjxi

��

Using bij � aijwj and a change in velocity yields the second order replicator equation with inter�

action matrix B�

Remark� A barycentric transformation shows that the replicator equation �R� has qualitatively

the same phase portraits since the constants �j are all positive�

Little can be said in general about the behavior for small translation rates� The extremal models

discussed in sections and � show that bifurcations leading away from replicator like� behavior

can occur for slow replication�

�Replication� Translation� and Mutation

Mutation is an integral component of any evolving system� Since the e�ect of mutation is to

introduce new� hitherto unknown species into the system it cannot be adequately modeled within

the framework of di�erential equations if the number of possible species is very large� Nevertheless�

it makes sense to consider the e�ect of mutations between the species in our reaction system�

Mutation can be viewed as a perturbation that pushes the �ow of the replicator equation away

from the boundary into the interior of the state space �� Assuming both incorrect replication

and incorrect translation we have to deal with the following reaction network

�A� �Xk � TlQkmakl�� Ik � Im � Tl

�B� � IkQTkmwk�� Ik � Tm

The accuracy of the replication step is described by the probabilities Qmk of forming a mutant Im

from a template Ik� Correspondingly� a translation product of type Tm is produced from a gene

Ik with probability QTmk� The kinetic equations read

�xk �nX

j��

Qkjxj

nXi��

ajiti � xk

�tk �nX

j��

QTkjwjxj � tk

T

� ��


where the �uxes and T are adjusted such thatP

j xj � � andP

j tj � �� The terms in the

above equation can be rearranged in order to emphasize the perturbational nature of mutation in

this model��xk � xk ��At�k � hx�Ati� �

Xj ��k

�Qkjxj�At�j �Qjkxk�At�k�

�tk � wkxk � tkhw� xi�Xj ��k

�QTkjwjxj � QT

jkwkxk� �RTM �

For sake of de�niteness we assume that the mutation matrix Q is of the form

Q �

�BB�� n � ��

� �� n� ��

��

� � � � � �� n � ��

�CCA

An analogous representation is assumed for QT � with a translation error rate � replacing the

mutation rate �� Then �RTM� simpli�es to

�xk � xk ��At�k � hx�Ati� � �P

j ��k �xj�At�j � xk�At�k��tk � wkxk � tkhw� xi � �

Pj ��k �wjxj �wkxk�

Let us use the notation � � �x� t� � Sn � Sndef

�� S� Then the replication�translation model with

mutation takes the form

�� RT �� M��# ��

where RT is the error�free replication�translation part andM describes the e�ects of mutation� For

the full model �RTM� even the computation of the equilibrium points can be done only numerically

except for a few special cases discussed below� The perturbation methods described in �� are

applicable here�

First we need some notation for the subsimplices that constitute the boundary of the state space

S� Let (FK � Sn be the subsimplices on which xk � � for all k � K� where K is an index set which

is neither f�� ng nor empty� The relative interior of the sub�simplex (FK will be denoted by

FK �

Denition �� A vector �eld M��# �� on S is a mutation �eld if it has the following properties

�i� M � �Mx�Mt� is continuously di�erentiable and kMk� is bounded on S � I � J � where I

and J are non�empty intervals I� J � IR��

�ii� M��# �� for all � � S�

�iii� Mxk��# �� if xk � � and M

tk��# �� if tk � � for all �� I � J� n f�� g�

We say that M is a strong mutation �eld if the weak inequalities in condition �iii� are replaced by

strong inequalities�

� ��


Lemma �� i� (FK � (FK is invariant under the ow of �RT�� ii� (FK � Sn is invariant under the

ow of �RTM� with � � � and arbitrary ��

Proof� �i� Let k � K� From xk � tk � � we infer immediately that �xk � �tk � � as well� thus

(Ffkg � (Ffkg is invariant� Repeating this argument for all k � K implies the proposition�

�ii� If � � � and � � � we �nd �tk � � if tk � � and xk � � �� xk � �� Thus (FK �Sn is invariant�

A direct analogue of the rest�point migration theorem which is proven in �� for selection�mutation

equations can be obtained for �RTM�� and even more general versions of models of erroneous

replication and translation� The proof of the following result parallels the selection�mutation

version line by line� therefore we omit the details here�

Theorem �� Let "� � (FK � (FK be a rest point of �RT�� let M��# �� be a strong mutation �eld�

Then there is an �� such that for all � � � �� and all � � � � �� the following statements

are true

�i� If the rest point "� is regular then �� intS if and only if the transversal eigenspace ET �"��

is stable� i�e�� if "� is saturated�

�ii� If the transversal eigenspace ET �"�� has at least one positive eigenvalue� say �l � �� then all

�xed points �j�� derived from "� lie outside the state space S�

a) b) c)

d) e) f)

Figure � The development of the �xed points of the competitive model for various values of �� where � � � for

n � �� a� no mutation� b� subcritical values for �x� c��f� supercritical value for �x� � �Sink� � � Saddle

� ��


The consequences of the rest point migration theorems are dramatic� Mutational perturbations

drive all non�saturated rest points into the non�physical exterior of the state space� The dynamics

of the systems are thus simpli�ed due to the reduction in the number of the �xed points� This is

the same behavior as has been observed for the selection mutation models in �� Figure shows

an exceptional case� all �xed points are saturated in this model and are therefore pushed into the

interior of the state space� The hypercyclic model discussed in section �� on the other hand� is left

with a unique rest point for small � � ��

Non�perturbative results can be obtained only for very simple interaction matrices� This matches

the situation for selection�mutation equations discussed in �� A few results can be obtained for

circulant interaction matrices A and equal translation rates wk � w for all k�

Lemma �� Suppose A �� is circulant and all translation rates are equal to w � �� Then

"� � �"x� "t� � �n�� is the unique interior equilibrium of �RTM��

Proof� Since A is circulant� �A� "t�j � � �j� Thus h"x�A"ti � �h"x��i � � and hw� "xi � w�

Substituting this into �RTM� immediately yields the lemma�

Let us now consider two special cases in more detail� The symmetric competitive model has

interaction matrix A � kI� The di�erential equation �RTM� reduces to

�xk � k � xk �tk � hx� ti� � � k

��n � ��xktk �X

j ��k

xjtj

��

�tk � w�xk � tk� � � ��n � �� wxk�

The eigenvalues at the interior equilibrium �n �� are readily computed� For the two external

eigenvalues we �nd &��x � �k n and &��t � �w� For the remaining eigenvalues one �nds

&j� ��k�� w

��

�

r�n � k��

kw

n�� n�� n��

The critical values of � and � for which the interior �xed point becomes stable are readily obtained

from the above expression�

�crit �n� � �

n�n� � ��and �crit �

�n��

n�n��

The stability of the unique interior rest point �n �� can be determined also for the symmetric

cooperation model with interaction matrix akj � �i�j�� The Jacobian has the same external

� ��


eigenvalues as in the case of the symmetric competition model discussed above� For the physically

relevant eigenvalue we �nd

&j� � �k� �w

��

�

r�w � k��

kw

n�� n�� n�� exp��

j

n�

The critical values of the mutation rate can be computed from the above equation as solutions of

a cubic equation� In the case n � � the analytical solution is tractable but too messy to be printed

here� With the help of Mathematica we could show that if �n�� is a stable equilibrium for � � ��

then it remains stable for arbitrary values of �� For higher dimensions numerical studies show that

the interior rest point becomes stable for large enough mutation rates�

The behavior of �RTM� is in general very similar to the behavior of a second order replicator

network under the in�uence of mutation� Increasing mutation rates force the �ow towards the

middle� of the state space� and very high mutation rates lead yield a globally stable interior

equilibrium ��

��Discussion

In this contribution we have compared the dynamics of the most simple replication�translation

model �RT� and the dynamics of second order replicator equations �R�� The version of �RT�

considered here is the simplest extension of the replicator equation model that separates the repli�

cating entities �genomes� from the catalysts mediating the replication process� We �nd that the

behavior of such a system is very closely related to the dynamics of an autocatalyic replication

process which does not involve translation� More precisely�

� The replication�translation model with rate constants aij for the replication of a genome of

type i using the catalytic activity of translation product j and with a translation rate wj

for the translation of a genome of type j corresponds to a second order replicator equation

�R� with interaction coe!cients bij � aijwj � We have shown that the replicator equation

with these interaction coe!cients is in fact the singular perturbation limit of �RT� for fast

translation�

� The dynamics of �R� and �RT� are also closely related for small translation rates� In

particular� the there is an interior rest point of �RT� if and only if the corresponding replicator

� � �


equation has an interior equilibrium� Furthermore� a rest point of �RT� is saturated if and

only if the corresponding equilibrium of �R� is saturated�

� The dynamics of a two�species system is the same in both models�

� Mutations a�ect both models in qualitatively the same way�

Not surprisingly� non�generic features of second�order replicator equations are not shared by the

replication�translation model� For instance� we �nd super�critical Hopf�bifurcation in three�species

models which do not occur in second order replicator equations� It is interesting to note in this

context that a monotonic distortion of the linear�response functions �Ax�k in a the second order

replicator equation can have the same e�ect ��

Of course� our model of translation is unrealistically simple� it does not consider the formation

of a complicated translational apparatus such as a ribosome� nor does it contain any regulatory

mechanism� But then� we have never claimed to model a realistic early translation mechanism��

The purpose of this contribution is to show that networks that are based on replication can be

�roughly� described by replicator equations even if the replicating entities� in our case the genotypes

together with their translation products� have non�trivial internal structure� A similar question

was raised in �� for quite di�erent variations of the replicator theme�

This investigation is thus only a �rst step towards a much more ambitious goal� Given a chemical

reaction network can we devise a coarse grained description in which the players are collections of

chemical species� such that the dynamics of the original reaction network is well simulated by a

dynamics on the coarse grained level�

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Documents

Autocatalytic Replication in a CSTR and Constant Organization · CSTR and Constant Organization Robert Happel Peter F. Stadler SFI WORKING PAPER: 1995-07-062 ... a mathematical theory