Evolution on RandomStructuresChristian M. ReidysSimon M. Fraser

SFI WORKING PAPER: 1996-11-082

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SANTA FE INSTITUTE

�

Evolution on Random Structures

By

Christian M� Reidysa�b and Simon M� Fraserb

a Los Alamos National LaboratoryTSA�DO�SA� ���� New Mexico

bSanta Fe Institute��� Hyde Park Rd�� Santa Fe� NM ������ USA

�Mailing Address bSanta Fe Institute

��� Hyde Park Rd�� Santa Fe� NM ������ USAPhone �� ����� ������� Fax �� ����� ��������

E�Mail fduck� smfrg�santafe�edu

� � INTRODUCTION

Abstract� In this paper we investigate the relation between the structure and dynamicsof molecules� using a level of coarse graining at which we consider a molecular structure asa �random structure�� A random structure consists of �i� a �random� contact graph and�ii� a family of relations imposed on its adjacent vertices� The vertex set of the contactgraph is simply the set of all indices of a sequence� and its edges are obtained by pickingsecondary and tertiary bonds �from the set of all possible bonds� in two randomizationprocedures� The corresponding relations associated with the edges are viewed as secondarybase pairing rules and tertiary interaction rules respectively� Mappings of sequences intorandom structures are constructed� Here� the set of all sequences that map into a particularrandom structure is modeled as a random graph in the sequence space� the so called neutralnetwork� We analyze the graph structure of the contact graphs of random structures andtheir union� and show how their graph theoretic properties in�uence the dynamics ofsequences mapping into them� In particular� we see a phase transition �in the limit of longsequences� in the union graph� which is manifested in the emergence of a giant component�The critical parameter for this phase transition is the fraction of tertiary interactions inthe molecule� A replication�deletion experiment reveals that this dramatic change inmolecular structure has signi�cant e�ects on the dynamics of the optimization process�This results in a non�linear relation between the fraction of tertiary interactions in thebiomolecules� and the times taken for a population of sequences to �nd a high��tnesstarget structure� These results have important implications for evolutionary optimizationin biopolymers� in particular the evolution of viruses�

Key words� random structure� sequence�structure mapping� random graph� connectivity�giant component� hitting time� optimization

� Introduction

The folding of molecular sequences into their spatial structures is of central interest inbiophysics� Its properties are important for the understanding of evolutionary optimizationof biopolymers and the theory of molecular evolution� Obviously� the term �structure�can re�ect di�erent levels of coarse graining� In biophysics �structure� is de�ned in termsof some physical conditions� for example minimum free energy or kinetic parameters��Structure� can also be de�ned as the set of all a�ne coordinates of the atoms of a moleculeor it can consist of a list of all pairs of coordinates of the sequence that are paired by meansof chemical bonds� In this paper we will consider �structure� as such a correlation scheme�In particular we will not assume that this scheme has to ful�ll constraints that might arisefrom an embedding in the three dimensional space� A structure will consist of a graph anda multi�set of relations imposed on the extremities of its edges� Its vertices correspond tothe indices of the nucleotides and its edges correspond to the bonds�

Intensive research has been done in computing the minimum free energy structure of aparticular RNA sequence ��� ��� �� focusing on the question of how to determine themost likely structure for a given sequence� Schuster and coworkers �� provided an impor�tant change of perspectives by concentrating on the mapping from sequences into RNAsecondary structures� in the context of molecular evolutionary biology� These mappingshave been used to induce �tness landscapes in which evolutionary optimization has beenstudied� In those landscapes� for example� the lengths of neutral walks� i�e� paths wheresubsequent sequences have a Hamming distance one and all sequences map into a �xedsecondary structure� have been analyzed ��� �� ���� It has been observed that in case ofAUGC�sequences one can walk almost into maximal Hamming�distance without changingthe structure�

In the last two years a mathematical model for the mapping into RNA secondary structureswas developed ��� ���� which changed the perspective towards graph theoretic propertiesof the sequences that map into a �xed structure� These sequences have been shown toform neutral networks in sequence space �being neutral on the level of structures�� Neutralnetworks themselves can be modeled as random graphs and exhibit threshold values fortheir density and connectivity properties� For biologically�realistic parameters� the theoryimplies the existence of dense and connected networks that percolate sequence space andare thereby well suited to searching for new structures� Simultaneously� exhaustive map�pings using biophysical folding algorithms have been performed �� �� in order to comparethe random graph model with biophysical maps� In case of GC�sequences of length ��this work veri�ed the existence of neutral networks which decomposed into a few �mostly��� or � components� Furthermore� random graph maps have been studied ����� whichhave similar structure� Thus� there are robust features of mappings into RNA secondarystructures ����� which depend more on the combinatorial properties of the molecules� ratherthan particular physical parameters �for example� the strength of certain base pairs��

So far all results have been obtained for a special class of structures� namely RNA sec�ondary structures obeying Watson�Crick base�pairing rules� However� the above resultsimply ����� that molecular structures can be studied from a rather new point of view theyare correlation schemes that induce landscapes that can be searched e�ectively by pointmutations� In this context� some combinatorial features of bonds in biomolecules are ofcentral importance� In a biomolecule there are at a �rst approximation two classes ofchemical bonds that is secondary and tertiary ones� Secondary bonds impose a unique�ness constraint for the involved nucleotides�no base can have more than one secondaryinteraction� In addition� for secondary bonds there exists a context independent base pair�ing rule which determines which pairs of nucleotides are able to establish bonds� Tertiarybonds are known to have context dependent rules and in general do not ful�ll a unique�ness constraint as described above for secondary bonds� In order to generalize the RNArandom graph model to mappings in D�RNA or proteins� random structures have beenintroduced �����

� INTRODUCTION

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Random structurerealised by a sequence:

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Contact graph: Components:

Figure � A contact graph� consisting of an ordered set of vertices �numbered�� between

which there can be either secondary �gray� or tertiary �thin black� edges� together with its

set of components� On the right hand side� the bases of one compatible sequence are shown

on the vertices� indicating that� in the random structure� certain relations associated with

the edges have to be ful�lled �in this case Watson�Crick base�pairing rules��

A random structure consists of the following pieces of data �i� a random contact graph��nm�c�

� and �ii� a family of relations on adjacent vertices� Here� m is the number of sec�ondary bonds� and c� the fraction of nucleotides involved in tertiary interactions� Thecontact graph is a graph on the n indices of the coordinates of a sequence� whose edgeset is the union of the edge sets of two random graphs� The �rst is a ��regular graph on�m vertices� obtained by picking m pairs of indices without replacement� The second isa random graph on n vertices� obtained by selecting the remaining

�n�

� �m edges withindependent probability p � c�

n� Here� c� is the probability of a speci�c nucleotide being

involved in a tertiary interaction� It has been shown in ���� that in the limit of long se�quences almost all vertices of the random contact graphs are contained in tree componentsof logarithmic size �relative to sequence length��

Random structures are thus robust with respect to point mutations� This robustnessis a well known factor in molecular biology and has been discussed in the context ofneutral evolution ���� For example let us consider a point mutation in a nucleotide ofa component� According to the relations �rules� associated with the edges �bonds�� ahigh fraction of all nucleotides of this component has to be changed in order to staycompatible� The probability of the sequence remaining compatible with the structuredecreases exponentially with the size of the component in which the mutation occurred�Hence contact graphs with small components are robust with respect to point mutations�

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Figure � This �gure illustrates the notion of compatibility of a sequence with respect to

a structure� For three types of components �single nucleotides a� nucleotides in secondary

interactions b� and those in secondary and tertiary interactions c�� compatible sets of

vertices are shown�

One central question for the biomolecular optimization process is how well the set ofstructures is searched by point�mutations� In this context� the structure of the unionof two contact graphs �and of biomolecules in general� is of particular importance� Forrandom structures� we can determine the structure of this graph� and show its in�uence onthe dynamics of the optimization process� Here we describe our basic experiment� Let snbe a random structure that has intermediate �tness� and has a high fraction of a population

of sequences on its neutral network� Now suppose that s�n is a target structure having ahigher �tness� Then we would like to know how likely it is that sequences folding into snwill be mutated into sequences folding into the target structure� If we take the union of allbonds of the contact graphs of sn and s�n we obtain a new graph� It is exactly this uniongraph that will allow to describe how �close� the above two structures come� Formallywe could view the union graph as a new type of contact graph� We could then use this todetermine sequences that ful�ll the constraints imposed by both the constituent randomstructures� Now� if on the one hand the union graph decomposes in small components�the above arguments discussed in relation to single contact graphs would apply it ishighly likely that �bi�compatible� sequences exist� which are capable of folding into bothstructures� On the other hand if the union graph exhibits a large component of ordern it is unlikely that bi�compatible sequences will exist� In this component there willbe many cycles which impose large numbers of constraints that can never be ful�lledsimultaneously by a sequence� Therefore it will be di�cult to switch between both sn andthe target structure s�n by point mutations� We will show how the dynamics of this searchprocess change in relation to the m and c� parameters�

� � RANDOM STRUCTURES

The paper is structured as follows �rst we introduce the concept of random structuresand give some statistics on their graph theoretic properties� Second� we study the relationbetween the structure of the contact graph and the dynamics of evolution on these randomstructures�

� Random structures

��� Terminology

A graph X consists of a tuple �vX� eX� and a map o� t eX �� vX � vX� vX is calledthe vertex set and eX the edge set� An element P � X is called a vertex of X an elementy � X is called an edge� The vertex o�y� is called the origin of y and the vertex t�y�is called the terminus of y o�y�� t�y� are called the extremities of some edge y� There isan obvious notion of Y being a subgraph of X� We call a subgraph Y induced� if for anyP� P � � Y being extremities of an edge y � X� it follows y � Y � A path in X is a sequence�Q�� y�� Q�� y�� � � � � yn� Qn���� where Qi � X� yi � X� o�yi� � Qi and t�yi� � Qi��� Apath such that Q� � Qn�� is called a cycle� X is called connected if any two vertices arevertices of a path of X� A connected graph without cycles is called a tree� Being connectedis an equivalence relation in X� and the maximal connected subsets of vertices are calledcomponents of X� If we let Y be a subgraph of X� then the closure of Y in X� Y � is theinduced subgraph of all vertices of X that are adjacent to some vertices of Y � A subgraphY � X is called dense if and only if Y � X� Finally� a vertex P is called isolated in X ifit is not an extremity to an edge y � X�

We now introduce the contact graph of a random structure� For this purpose� let � � c� � �and � � c� � � be positive constants and n be the sequence length� Suppose that m�n�is a natural number such that �m�n��n is a monotonically increasing sequence with limitlimn��

�mn

� c�� m corresponds to the number of secondary bonds of the molecule�We will write a sequence V � Qn

� as V � �P�� � � � � Pn�� Qn� is a generalized n�cube whose

vertices are sequences of length n over the alphabet A of size �� and in which two sequencesare adjacent if they di�er in exactly one coordinate�

Let now X� be a partial ��factor graph on �m indices� say� f�i� � � � � � �i�mg � f�� � � � � ng� X�

is the contact graph formed by all secondary interactions� Next let X� be the random graphobtained by selecting all possible edges between the n nucleotides except the secondaryedges with probability c��n� Clearly� the graphs X� form a �nite probability space byassigning to each ��regular graph uniform probability� Analogously� the graphs X� form a

�nite probability space where a graph with k edges has probability pk��� p��n

���m�k withp � c��n ���� The graphs X��X� induce the graph X� �X� whose vertex set is f�� � � � � ngand whose edge set e�X� �X�� is the �disjoint� union of all X��X��edges� X� � X� iscalled the contact graph� The probability space formed by the graphs X� � X� will be

��� Statistics of random structures �

referred to as �nm�c�

�

A random structure� sn� on n nucleotides of a �nite alphabet A consists of the followingpieces of data

a contact graph X� �X�

a family of symmetric relations �R��Ry�y�X�� where R��Ry � A�A�

Each Ry is supposed to have the property for all a � A there exists one b � A withthe property aRyb� The relation R� is motivated by Watson�Crick base�pairing rules

observed in RNA secondary�structures� For y � X� the relation Ry corresponds to aspeci�c �tertiary� interaction rule that might be context dependent�

A vertex �sequence� V � Qn� is called compatible to sn if and only if

for all bonds y of the partial ��factor graph X� its nucleotides indexed by the extrem�ities fo�y�� t�y�g have the property Po�y�R�Pt�y� �note that since R� is symmetric wealso have Pt�y�R�Po�y��

its nucleotides ful�ll for all tertiary bonds y � X� Po�y�RyPt�y��

The set of compatible vertices with respect to the random structure sn is called C�sn��By construction there are n � �m vertices not incident to an X��edge and there areasymptotically �n� �m�e�c� isolated vertices in X� �X��

��� Statistics of random structures

We already discussed the relation between the contact graph and the robustness of therandom structure with respect to point mutations� In this section we will analyze in detailthe graph structure of �i� the contact graph and �ii� of the union of two contact graphs� Anumber of asymptotic results on contact graphs and their union has been proven in �����As is typical for random graph results� these hold in the limit of long sequences� Here wewill verify the extent to which the results apply for sequences of �nite length�

First the expected number of paths of length � in contact graphs is given by

n�c� ! c��� � ���

This implies that the contact graphs decompose with probability � into small components�For this purpose we give some bounds for c�� c� that are observed in RNA and protein

� � RANDOM STRUCTURES

structures �� c� ��� and � c� ���� Plugging this into eq���� we obtainimmediately that most nucleotides of the contact graph �X� �X�� are contained in smallcomponents� We further compute the probability of having a mutation in a component ofsize � in the contact graph see Figure a�

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c2 = 0.15

a) b)

Figure a� The distribution of vertices in components of size � in a single contact graph�

for dierent values of c� �the fraction of tertiary bonds�� b� The scaling of the largest

component in a single contact graph with sequence length �number of vertices n� for c� ����� and ����� Also shown are �tted lines with y � Cln�n��n� with C � ��� and ��respectively�

In addition we have also analyzed how the largest component in the contact graph scaleswith the sequence length �Figure b�� According to ���� the largest component is at mostof the order C ln�n� with constant C � � for c� � ����� The above data illustrate� however�that the largest component has in fact size C ln�n� and we have computed values of Cfor two di�erent c� parameters �legend to Figure �� We complete our analysis of contactgraphs by computing the distribution of the numbers of cycles in �X� � X��� The dataclearly indicate that cycles are extremely rare events in contact graphs�see Figure �a�

Next we turn to the structure of the union graph� Following ���� there is a phase transitionin �X� �X �

��� �X� �X �

�� concerning the emergence of a largest component of order C �nwith constant C � � �� The latter transition is expressed in terms of c�� the average numberof tertiary interactions per nucleotide and c�� the fraction of secondary interactions� Theexact result reads that for small values of c� the components of the union graph have sizesbounded by C ln�n� and� in the limit of large sequence length� for c�� c� such that

�c���� c��c� � � � ���

there exists a unique large component of size Kn� K � �� with probability tending to

��� Mapping in random structures �

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a) b)

Figure a� The size of the largest component in the union of two contact structures

for c� � ��� as a function of c�� Curves are shown for graphs with ������� nodes with

� � c� � ���� and� inset� for a graph of ���� nodes with c� ���� b� The size of the

largest component as a function of sequence length n� Curves for various values of c� are

displayed�

�� We thus compute the fraction of nucleotides contained in the largest component of�X� � X �

�� � �X� � X �

�� and the order of the size of the largest component� see Figure� We next consider the so called sequence of components of �X� � X �

�� � �X� � X �

���This is the multi�set of the sizes of its components ordered by size� For increasing c� thelarge component engulfs the other components� as shown in Figure �� Moreover� since thelargest component is of size C �n� it contains most cycles of the union graph� To illustratethis we computed the distribution of the number of cycles in �X� �X �

�� � �X� �X �

�� forvarious c� parameters� the results of which are shown in Figure �b�

��� Mapping in random structures

In order to study the dynamics of the evolutionary optimization process� we have to es�tablish a mapping between sequences and random structures ����� If we �x a randomstructure sn� the preimage is necessarily contained in C�sn�� the set of compatible se�quences� As we have shown in the previous section� the underlying contact graph hasalmost all vertices in tree components of at most logarithmic size� This contact graphinduces a partition of the indices f�� � � � � ng into its components �Figure ��� Accordingly�we regroup the indices of the nucleotides of a compatible sequence into the components ofthe contact graph� Formally we can now consider each multi�set �Pi� � � � � � Pik�� consistingof nucleotides whose indices belong to a component of the contact graph� to be an element

�� � RANDOM STRUCTURES

0.0001

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0 100 200 300 400 500 600 700

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Component size l

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od

e in

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om

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ze l)

Figure � The sequence of components in the union of two contact structures for c� � ����with c� � ����� ����� ����� Curves are shown for graphs with ���� nodes data are mean

probabilities from ���� randomly constructed graphs�

of a new alphabet� Ak� Accordingly we can rewrite a compatible sequence as �Ai� � � � � � Ai���� being the number of components of the contact graph��

To illustrate this let us consider an RNA secondary structure with respect to the biophysi�cal alphabet A�U�G�C� The latter has a contact graph whose edges are exactly the pairedpositions f�i�� ik�� � � � � �im� ij�g� These are also all nontrivial components of the contactgraph� The Watson�Crick base pairing rule� AU�UA�GC�CG�GU�UG� corresponds tothe induced alphabet� Accordingly� a compatible sequence �P�� � � � � Pn� can be rewrittenas �Pi� � � � � � Pik � �Pj� � Pj��� � � � � �Pjr � Pjr���� where each pair �Pjk � Pjk��� ful�lls the Watson�Crick base pairing rules� The set of compatible sequences is the vertex set of Qn��

� �Q���

where � is the number of base pairs�

In general the set of compatible sequences is the vertex set of

hYi��

Qni�i

whereXi

i � ni � n �

�i � jAij� h is the number of components� and ni the length of the i�th component ofthe contact graph� Next we construct the preimage of the random structure sn� It willbe a random induced graph by selecting the vertices in each factor Qni

�iwith independent

��� Mapping in random structures ��

a) Single contact graph

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b) Union of two contact graphs

Number of cycles l

Figure � The probabilities of obtaining structures with cycles� in a� single contact graph�

and b� the union of two contact graphs� for dierent values of c��

probability i� Note that �vertex� here corresponds to a multi�set �Pi� � � � � � Pik� consistingof nucleotides whose indices belong to a component of the contact graph of sn� In thissense �vertex� can be viewed as a certain segment of the sequence� i �i being the index ofa component� can be interpreted as the stability of the random structure with respect to amutation that has �i� occurred in the i�th component and that has �ii� led to a compatiblesequence� To summarize� the preimage of a random structure is obtained by selecting cer�tain segments of sequences in Qn

� at random� For this process the mathematical structureof randomly induced subgraphs of generalized n�cubes is of particular relevance� It hasbeen shown ��� ��� that � � �� ���

p��� is a threshold value for density

limn��

nf�n is dense in Qn� g �

�� for � �

� for � �

and also a threshold value for the connectivity of random induced subgraphs of Qn�

limn��

nf�n is connectedg �

�� for � �� ���

p���

� for � �� ���p��� �

Further results on random induced subgraphs can be found in ���� The above resultssuggest that the preimage of a random structure consists of large connected subgraphs insequence space� Finally� in order to de�ne a complete mapping into random structuresby the above procedure we only have to iterate the above process� We obtain mappingsf Qn

� �� fsng by constructing the corresponding preimages as random graphs as follows

�� � OPTIMIZATION ON RANDOM STRUCTURES

we �x a mapping r fsng � N having the property j i � r�sj� � r�si� and set

f��r �s�� � �n�s�� f��r �si� � �n�si� n�j�i

��n�si� � �n�sj �� �

� Optimization on random structures

In this section we will study the time evolution of �nite populations V of sequences whichare replicated nucleotide by nucleotide with some independent error probability p� Thereplication event itself is a point process more precisely a birth�death process in whichsequences are chosen for replication with respect to their �tness� and randomly for deletion�Our main objective is to determine the time for a population to �nd the neutral network ofa given target structure� the so�called hitting time� depending on the fraction of nucleotidesinvolved in tertiary interactions�

Let � Qn� � fsng be a �xed sequence to structure map� and let f fsng � R be a

mapping assigning a �tness value to each structure� We obtain� taking the compositum ofour maps� a �tness landscape on the sequences

Qn� �� fsng �� R V �� ��V � � sn �� f�sn��

��� The replication�deletion process

A population V� of size N � is a ��nite� multi�set of sequences �Vi j i � N� where fVi j i �N g � Qn

� and N � �� The theory of point processes provides a powerful tool by identifying�Vi j i � N� with an integer valued measure � Qn

� � R�

V � �Vi j i � N� �� � �

NXi��

gVi � where gVi�v� �

�� forV �� Vi� otherwise �

We call the set of sequences where � is nonzero the support of �� Clearly� the restriction

of � to a subgraph Y � Qn� corresponds to considering subpopulations on the vertices of

Y � Note that ��f���s�� is the number of elements of V contained in f���s��

The time evolution of � is then obtained by a mapping from �Vi j i � N� to the family�V �i j i � N� as follows we select an ordered pair �Vl� Vk� where Vl� Vk � fVi j i � N g�For this purpose let ress� be the restriction of � to all sequences that are mapped intos� Clearly the subpopulation that corresponds to ress� consists of sequences all having�tness f�s�� Accordingly the average �tness of � reads

f� �Xs

��f���s��f�s� �

��� Hitting times �

Now� the �rst coordinate Vl of the above ordered pair is chosen with probability f�sVl��f�among the elements of V� The second coordinate of the above pair is selected withuniform probability on �Vi �� Vl j i � N�� i�e� ���N � ��� We select those pairs of sequencesat equidistant time steps� and for a population of size N we refer to a generation as Nsuch time steps�

Next we map the �rst sequence� Vl � �x�� ���� xn�� into the sequence V � � �x��� ���� x�

n�� Thisis performed by assigning to each coordinate xi a x�i �� xi with probability p where allx�i �� xi are equally distributed and leave the coordinate �xed otherwise� This randommapping l �� v� is called replication� Finally� we delete the second coordinate of the pair�Vl� Vk�� that is Vk and have a mapping �Vl� Vk� �� �Vl� V

��� Thereby we obtain a newfamily by substituting the Vk by the V �� This process is referred to as the replication�

deletion process�

��� Hitting times

Let us describe our basic experiment� First� for a particular value of c� we select two

random structures �s�c��n � s�n

�c���� whose contact graphs �X� �X�� X�

��X �

�� are cycle�free�and have underlying c� � ���� This biologically�realistic value is taken from data on longRNA sequences� in which about ��" of the nucleotides are in Watson�Crick base pairs���� ��� We have chosen c� � �� ���� ���� ����� ���� ����� The mapping f fsng � R is

de�ned as follows� We assign to s�c��n the �tness ��� and to s�n

�c�� ����� independent of c� all the other structures have �tness ��

Second� we construct the neutral networks with respect to s�c��n and s�n

�c��� For any se�

quence that is compatible to both structures� s�c��n and s�n

�c��� we �rst randomly selectone of the ordered pairs �sn� s

�

n� �s�n� sn�� Then we use successive experiments with anindicator random variable� X� where �X � �� � �biased coin �ips�� to determine intowhich structure a bi�compatible sequence is mapped� In case of X � � in the �rst coin��ipwe map the sequence to the �rst coordinate of the selected pair and are done� In caseof X � � we �ip again� and� for X � � we map the sequence to the second coordinate�otherwise we assign a �tness value of � to the sequence� Accordingly� we obtain two neu�

tral networks� � � �n�s�c��n ���� � �n�s�n

�c��� which induce a random landscape with three�tness values

f��P�� � � � � Pn�� �

���

� for �P�� � � � � Pn� �� �� � ������ for �P�� � � � � Pn� � ����� otherwise

Third� we initialize the replication�deletion process� as described in section ��� by gen�

erating N sequences at random to be on the neutral network of s�c��n � We �x the single

digit error probability p such that pn � � and study the time evolution of this population�The experiments are run until the neutral network of the target structure is found� or a

� � DISCUSSION

c2

0.0001

0.001

0.01

0.1

1

100 1000 10000

c2 = 0

c2 = 0.1

c2 = 0.15

c2 = 0.25

Hitting time (generations)

Pro

po

rtio

n o

f ru

ns

Mea

n t

ran

siti

on

tim

e (g

ener

atio

ns)

a) b)

0 0.05 0.1 0.15 0.2 0.250

500

1000

1500

2000

2500

Figure � a� The mean transition times for populations moving from the intermediate to

the high��tness structure for various c� values� n � �� population size ��� Data show

mean of ���� runs� which were terminated after ���� ��� generations� In all runs the

target structure was found� except in ��" of the runs with c� � ����� Bars are standarderrors� b� The distributions of hitting times for the same runs�

predetermined number of generations have elapsed �which is high enough that the targetstructure is found in the majority of runs��

The results of these replication�deletion experiments� shown in Figures � and �� clearlyindicate the dramatic e�ect of increasing c� values on the ability of searching populationsto �nd the target structure�

� Discussion

In this paper we have shown that even on the level of random structures there existsa signi�cant relation between structure and dynamics� We have considered moleculesconstructed at random according to very simple rules� which allow for evolutionary opti�mization� In particular� we have shown a dramatic change in the optimization performancein relation to the number of chemical bonds in the molecule�

In some sense random structures are generalizations of biomolecular secondary structures�which are formally planar knot�free graphs together with rules associated with their bonds�Knot�freeness� here� requires that edges in the planar graph do not intersect� i�e� for anytwo edges �i� k��r� l� we have either i r and k � l� or� r i and l � k� Random

��

0.05 0.1 0.15 0.2 0.250

50

100

150

200

Mea

n h

itti

ng

tim

e(g

ener

atio

ns ×

10-3

)

c2

10055

25

21

23

Figure � For sequences of length ��� and for various c� values� the graph shows the

mean transition times for populations of ���� sequences� moving from the intermediate

to the high��tness structure� Runs were terminated after ��� ��� generations� the lowest

completion rate being at c� � ���� where the target structure was found in only ����" ofthe runs� Means include data from non�complete runs� Bars are standard errors �gures

above the columns are the total number of runs�

structures� as secondary structures ������� induce neutral networks in sequence space� i�e�extended� mostly connected subgraphs consisting of all sequences that are all mapped intothe random structure� Those networks consist of a few components� whose size dependson the fraction of neutral point mutants �with respect to the structure�� They are stableunder point mutations and hence allow for neutral evolution ����

However� random structures di�er from secondary structures in two important regards�First� they may include tertiary interactions� and secondly� they need not satisfy such aknot�freeness condition� Although knot�freeness implies a number of inductive relations onsecondary structures of n�nucleotides ����� it complicates any probabilistic model of struc�tures because of additional conditional probabilities� Such a probabilistic model allowsthe graph structure of the union of two contact graphs �obtained by superimposing thebonds of two random structures� to be determined� and this contains essential informationon how close the corresponding neutral networks come in sequence space�

We have shown in the �rst section that the random graph results of ����� proven in the limitof in�nite sequences� already apply for relatively short sequence lengths� Contact graphsof random structures and secondary structures are similar in that they decompose intocomponents of small sizes� However� the properties of the union of two contact graphs ofrandom structures depend heavily on the fraction of tertiary interactions� For a c� � ���

�� � DISCUSSION

0 0.05 0.1 0.15 0.2 0.250

50

100

150

200

250

c2

Mea

n h

itti

ng

tim

e (1

03 g

ener

atio

ns)

Figure � The mean hitting times for populations of �� sequences �n � �� under two

dierent experimental regimes� The solid bars show runs as described in section ���� in

which populations may �nd the target structure via alternative� low��tness structures� The

open bars show runs in which the searching population was restricted to the neutral networkof the intermediate structure �see text�� Bars are standard errors of the means of ���

replicate runs�

the critical fraction of tertiary interactions for the emergence of a giant component inthe union graph of two random structures has c� � ���� as a upper bound �eq� �����For c� � � a lower bound on the critical fraction would be ������ The existence of thisdramatic change �Figure a�� which is a phase transition in the limit of long sequences�does not depend on c� as long as c� � �� Known D�structures �for example t�RNA�have values of c� which are well below this critical threshold� with about ��" nucleotidesinvolved in tertiary interactions�

We presented data in section �� which show how the changing structure of the uniongraph� with increasing c� values� in�uences the ability of a population of sequences to�nd a target structure� The data show a highly non�linear response in hitting timeswith increasing c�� For example� for n � ��� a change from � to ��" of tertiary bondsincreases the hitting time ���fold� from ���� to ����� generations �Figure ��� Becauseof computational limitations� experiments could only be performed for relatively shortsequences� However� the emergence of the giant component in the union graph becomesmore sudden for longer sequences� and we thus expect our results to become even moresigni�cant for higher values of n�

REFERENCES ��

In this context it is interesting to consider an experiment in which searching populationswere restricted to the network of the intermediate structure� More explicitly� sequencesmutating into low��tness structures were replaced by sequences generated at random tobe compatible with the structure of intermediate �tness� Results of such an experimentfor n � � are shown in Figure �� Here we observe signi�cantly longer hitting times�but the non�linear relation with c� still holds� Relating these two experiments allowsinsight into the searching of structure space by evolving sequences� In general� neutralmutations are exactly the driving force for evolutionary search they allow a populationto explore sequence space by moving on the extended neutral network� The signi�cantdi�erence in the hitting times of the basic experiment and that shown in Figure � indicatesthat structure space is searched largely by the mutants that �fall o�� the network of theintermediate structure� However� for low c� parameters� it is possible for a populationto switch directly between neutral networks� In fact� such transitions were observed inthose experiments where no element of the population was able to leave the network of theintermediate �tness structure� That is� sequences of intermediate �tness mutated directlyinto the sequences realizing the target structure� This extreme case is approached inour standard experiment by vastly increasing the �tnesses of the intermediate and targetstructures �������

In summary� we have veri�ed the theoretical results of ���� regarding the properties ofsingle contact graphs and their union for sequences of �nite length� Our experimentalresults show how the fraction of nucleotides in tertiary interactions alters dramatically thestructure of the union of two contact graphs� and the e�ect of this changing structure onthe ability of populations of sequences to move from the neutral network of one structure tothat of another� More generally� we show that a coarse�graining of molecular structure atthe level of contact graphs is a useful approach� and enables the prediction of biologicallyimportant properties of biomolecules� such as the frequency of tertiary interactions�

Acknowledgments We want to thank Christopher L� Barrett for stimulating discussions�and Rob Farber and Bill Reynolds for generous donation of CPU time� SF is funded byDARPA under grant ONR N��������������

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