Approximate Scaling Properties of RNA Free Energy Landscapes...Baskaran et al Scaling in RNA...

Approximate Scaling Propertiesof RNA Free EnergyLandscapesSubbiah BaskaranPeter F. StadlerPeter Schuster

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SANTA FE INSTITUTE

Approximate Scaling Properties of

RNA Free Energy Landscapes

By

Subbiah Baskarana�b�c� Peter F� Stadlerb�d� and Peter Schusterb�c�d��

aES�� Marshall Space Flight Center� Huntsville� AL�� USAbInstitut f�ur Theoretische Chemie� Universit�at Wien� Vienna� Austria

cInstitut f�ur Molekulare Biotechnologie eV� Jena� GermanydSanta Fe Institute� Santa Fe� NM �� USA

�Correspondence to Institut f�ur Theoretische Chemie� Universit�at Wien

W�ahringerstra�e �� A�� Vienna� AustriaPhone �� Fax ��

Email pks�tbiunivieacat

Baskaran et al� Scaling in RNA Landscapes

Abstract

RNA free energy landscapes are analyzed by means of �time�series� that are obtained fromrandom walks restricted to excursion sets� The power spectra� the scaling of the jump size dis�tribution� and the scaling of the curve length measured with di�erent yard stick lengths are usedto describe the structure of these �time�series�� Although they are stationary by construction�we �nd that their local behavior is consistent with both AR� and self�a�ne processes� Randomwalks con�ned to excursion sets i�e�� with the restriction that the �tness value exceeds a certainthreshold at each step� exhibit essentially the same statistics as free random walks�We �nd that an AR� time series is in general approximately self�a�ne on time scales up toapproximately the correlation length� We present an empirical relation between the correlationparameter � of the AR� model and the exponents characterizing self�a�nity�

Key Words

RNA Folding Excursion Sets Fractal Landscape AR� Process ��f noise

� � �


�� Introduction

Evolutionary optimization as well as combinatorial optimization take place on

landscapes resulting from mapping �micro�con�gurations to scalar quantities like

�tness values� energies� or costs �Schuster � Stadler� �� In most cases one

lacks a detailed understanding of the structure of �tness landscapes that underlies

a particular instance of biological evolution One resorts thus to using model land�

scapes Well studied examples include combinatorial optimization problems such

as the Traveling Salesman Problem� the Graph Matching Problem� or the Graph

Bipartitioning Problem� various spin glass models� among them the Sherrington�

Kirkpatrick models� and Kau�man�s Nk�models �Kau�man� �� The ruggedness

of a landscape� often measured by means of a correlation function� is of crucial im�

portance for the dynamics of the evolution process �Eigen et al�� Bonhoe�er

� Stadler� �� Detailed studies of the correlation structure of model landscapes

can be found� for instance� in the following references �Stadler � Schnabl� ��

Stadler � Happel� �� Stadler� �� Weinberger� ��a� Weinberger� ��b�

Weinberger � Stadler� ��

Exclusively in the case of RNA landscapes do we have a sound biophysical model

for the �tness function Models based on RNA secondary structure prediction

algorithms have been analyzed in great details in a series of papers �Fontana et al��

�� Fontana et al�� Bonhoe�er et al�� Tacker et al�� Schuster

et al�� Evolutionary dynamics on such landscapes was the topic of extensive

research as well �Fontana � Schuster� �� Fontana et al�� Huynen et al��

�� A detailed understanding of these landscapes is a necessary prerequisite

for building simpler models based on spin glass or Nk model landscapes that are

signi�cantly less costly in computer simulations and that lend themselves much

easier to analytical treatment

Weinberger �Weinberger� �� suggested to characterize a landscape by means of

a �time series� obtained by sampling the �tness values along a random walk in

sequence space While this method is rather indirect� it yields a data set that can

be analyzed by the standard methods of time series analysis �Hordijk� �� In

� �


this contribution we shall investigate the �fractal�like� features of landscapes in

terms of the approximate self�a�nity of these �time�series�

A great variety of systems� physical and biological� exhibit �� power spectra�

commonly called ��f�noise or � icker� noise Some examples are resistivity uctu�

ation in conducting materials �Weissman� �� luminosity uctuations of stars

and galaxies �Nolan et al�� ow uctuations of highway tra�c �Musha �

Higuchi� �� and of deep ocean waters �Taft et al�� frequency variations

of quartz oscillator �Attkinson et al�� the loudness uctuations in music and

speech �Voss � Clarke� �� In biological systems ��f noise has been reported

for nerve membranes �Verveen � Derkson� �� for the DNA sequences of the

non�coding introns �Voss� �� Li � Kaneko� �� as well as of coding regions

�Buldyrev et al�� In this paper we will show that the �time�series� sampled

along a random walk on a RNA free energy landscapes also leads to ��f noise

This contribution is organized as follows In section we review some notions that

are basic to the theory of �tness landscapes In particular� we introduce a variety

of correlation measures and highlight their relations with each other In particular

we consider the class of landscapes that lead to exponential correlation functions

of the �time series� obtained from simple random walks In section � we brie y

consider self�a�ne time�series and show that AR�� processes mimic self�a�nity on

time scales up to their correlation length These �nding are applied to free energy

landscapes of RNA in section � In particular� we shall see that the mountainous

parts of the landscapes do not di�er signi�cantly from the average �tness regime�

at least as long as the excursion sets do not fragment into tiny pieces Section �

concludes our discussion The relaxation time of a simple random walk on a

sequence space is computed in the appendix

� � �


�� Landscapes

��Rugged Landscapes

De�nition� A landscape is a map f C � IR� where C ! �X�d� is a �nite metric

space with metric d X �X � IR

In most applications of landscapes in biology� physics� or combinatorial optimiza�

tion the con�guration space �X�d� can be represented as a graph " Then two

con�gurations x and y are neighbors in " if d�x� y� ! � The metric d is often

obtained from an editing procedure that allows to interconvert two con�gurations

x� y � X by means of a �nite sequence of operations d�x� y� is commonly de�ned

as the number of operations in the shortest sequence that changes x into y or vice

versa In a biological context the �elementary operations� are in general muta�

tions We will restrict ourselves there to the case where X is a set of sequences

of common length n which are constructed from some alphabet with � letters In

this case d is the so�called Hamming distance �Hamming� �� and the graph "

is known a the sequence space Qn�� or Boolean hypercube in the special case � !

For a recent review see �Schuster � Stadler� �� Stadler� ��b�

��Correlation Functions

A very important characteristic of a landscape is its ruggedness Rugged land�

scapes are characterized by a large number of local optima �Palmer� �� the

fact that uphill walks are short and easily trapped in local optima� and by short

correlation lengths �Kau�man� �� There is ample evidence that heuristic op�

timization procedures work less e�ciently the more rugged a landscape is �Stadler

� � �


� Schnabl� �� Schuster � Stadler� �� It will be convenient to de�ne for a

given landscape f

f !�

jXj

Xx�X

f�x� ��f !�

jXj

Xx�X

�f�x� � f

��

It has been suggested by various authors �Eigen et al�� Fontana et al��

Sorkin� �� Weinberger� �� to measure �ruggedness� by some sort of corre�

lation measure We shall use the following de�nition� which was �rst proposed in

ref �Eigen et al��

��d� !�

��f�

�

jDdj

X�x�y��Dd

�f�x� � f ��f�y� � f

��

Here Dd denotes the set of all pair of vertices that have mutual distance d in the

graph " For a sequence space we have for instance

jDdj ! �n�� d�n

d

��

This de�nition is useful if " is a distance regular graph �Brouwer et al�� A

more general mathematical framework is developed in �Stadler� ��c� Happel �

Stadler� �� Stadler � Happel� ��

Weinberger �Weinberger� �� Weinberger� ��a� Weinberger� ��b� suggested

to investigate the properties of landscapes by sampling the values along a simple

random walk in the con�guration space C

x� � x� � x� � � � � � xk � � � �j j j jf� � f� � f� � � � � � fk � � � �

��

where xi and xi�� are neighbors in C At each step one of the neighbors of xi in "

is chosen with uniform probability� ie� the series fxig is a simple random walk on

C �Spitzer� �� By evaluating the con�gurations along the walk fxig we obtain

a random walk on the landscape� ie� the �time series� ffi ! f�xi�g This series

is stationary by construction

� � �


The autocorrelation function of a stationary time series is de�ned by

r�s� !hftft�si � hf�t i

��

where �� is the variance� which coincides with ��f de�ned above� and the angular

brackets indicate the expectation value taken over all random walks fxig and

all times t Provided the graph " is D�regular� ie� each con�guration x has

exactly D neighbors� we may write the transition matrix of the random walks as

T ! ��D�A The entry Axy of the adjacency matrix is � or �� depending on

whether the con�gurations x and y are neighbors or not It is shown in �Stadler�

��c� that the correlation function r�s� has the following algebraic representation

r�s� !�

��

hhf�Tsfi � f

�i� ��

Note that h � � � i denotes here a scalar product� not an expectation value# Another

useful �Fontana et al�� representation is

r�s� ! ��h �ft�s � ft�� i

��

The average squared di�erence h �ft�s � ft�� i was used as a correlation measure

in Sorkin�s pioneering paper �Sorkin� ��

The autocorrelation function ��d� of the landscape itself and the autocorrelation

r�s� of the �time�series� of the landscape are related via

r�s� !Xd

�sd��d� ��

where �sd is the probability that a simple random walk of length s ends at distance

d�x�� xs� ! d Explicit expressions for �sd can be found in ref �Fontana et al��

�� Stadler� ��a�� we shall not make use of them in this contribution

� � �


��Elementary Landscapes

For a quite large number of model landscapes it has been found that the corre�

lation function r�s� is exactly a decaying exponential �Stadler � Happel� ��

Weinberger � Stadler� �� numerically indistinguishable from a decaying ex�

ponential �Stadler � Schnabl� �� Stadler� �� or at least very close to a

decaying exponential �Weinberger� �� Weinberger� ��a� It has been argued

that a nearly exponential autocorrelation function r�s� would be generic for land�

scapes with a Gaussian distribution of �tness values �Weinberger� �� This

argument is wrong� however

It is not hard to check that r�s� is exponential whenever f is of the form f�x� !

f $ ��x�� where � is an eigenvector of the adjacency matrix A with eigenvalue

% Indeed� under these conditions one �nds r�s� ! �%�D�s In a more general

context is useful to assume that � is an eigenvector of the so�called graph Laplacian

�Mohar� �� for regular graphs we have & ! A �DE� where E is the identity

matrix� ie� the eigenvectors of A are the same as the eigenvectors of the Laplacian

& Landscapes of this type have been termed elementary Lov Grover �Grover�

�� found that a number of well known model landscapes are elementary� for

instance the landscape of the Traveling Salesman Problem In �Stadler� ��c� it

is also shown that r�s� is exponential if and only if the landscape is elementary

Note that the possible eigenvalues % are uniquely determined by the adjacency

matrix A� ie� by the geometry of the con�guration space As a consequence

there is only a �nite small number of possible values for the parameter def

��%�D

of the exponential decay� ie� it is not possible to construct a landscape f with

autocorrelation function r�s� ! s with an arbitrarily prescribed parameter '

in contrast to the case of merely constructing a time series In other worlds� only

a very special set of time series is generated by random walks on landscapes

� � �


��Power Spectra

Instead of a correlation function one can use power spectrum

S�� def

�� limN��

�

N

��

NXt��

ft cos��t�

$

�NXt��

ft sin��t�

� � ��

of the time series fftg as a means of characterizing the landscape Here N is

the number of points sampled from the time series fftg Power spectrum and

autocorrelation function of a stationary process are related by theWiener�Khinchin

theorem �see� eg� �Yaglom� ��

r�s� !

��

Z �

�

S�� cos��s�d�

S�� !��

�� $

�Xs��

r�s� cos��s�

��

A negative slope of S�� implies some degree of correlation in ft A steeper slope

implies a higher degree of correlation A signal fftg is called ��f noise if a log�log

plot of the power spectrum versus frequency can be approximated by straight line

with slope close to �� in the frequency range of interest More generally� one

speaks of ��fa noise if the slope is �a We shall return to this type of time series

in section �

The most common de�nition of a correlation length in physics is simply the integral

of the autocorrelation function In the discrete case it is convenient to use

(� def

��

$

�Xs��

r�s� � ��

Comparing this de�nition with the Wiener�Khinchin theorem� equ�� yields

the simple relation

S�� !��

(� ��

which can be used as an alternative way of estimating the correlation length of a

time series

� � �


��Excursion Sets

The parts of the landscape in which the values are close to the global maximum

or minimum are particular interest One might ask� for instance� how the �good�

solutions are distributed in sequence space) Are they clustered around a globally

optimal solution� or are con�gurations with close�to�optimal values scattered all

over the con�guration space) A suitable mathematical framework for this type of

questions is set by the notion of excursion sets �Adler� �� In this subsection

we collect a few de�nitions and their immediate corrolaries which will be useful

for the discussion of the RNA free energy landscapes in section �

De�nition� Let f X � IR be an arbitrary landscape

�i� A con�guration x is a local optimum if for all neighbors y of x holds f�x� �

f�y� Two con�gurations x and y are called neutral if f�x� ! f�y�

�ii� The set AE ! fx � Xjf�x� � Eg is called the excursion set of f at level E

A connected component of AE is called a cycle �Freidlin � Wentzell� ��

�iii� A connected subgraph B � A is called neutral network in X if all elements

are neutral� and if all neutral neighbors of any x � B are elements of B as

well

For su�ciently small E we have of course AE ! X� the entire con�guration space

On the other hand� if E is larger than the global optimum of f � then AE is

empty Clearly� E � E� implies AE AE� � hence excursion sets introduce a

hierarchical structure on the landscape In general� AE will not be connected� ie�

it will decompose into more than one cycle �connected component� Bounds on

the number of cycles can be obtained for elementary landscapes and the special

value E ! f � for details see �Stadler� ��c� Cycles play a prominent role in

the analysis of simulated annealing techniques on combinatory landscapes� see

�Azencott� �� for a recent review

Excursion sets� local optima� and neutral networks are closely related We list

here only a few simple geometric relationships �i� Suppose CE is a cycle and B

is a neutral network� then CE and B are either disjoint or B is subset of CE �ii�

� � �


Each cycle CE contains at least one local optimum �iii� A neutral network B is

a cycle if and only if it consists entirely of local optima Each cycle CE contains

a cycle of this type �iv� A neutral network which is a cycle contains no other

cycles except for itself �v� If a cycle consists of only one con�guration then this

con�guration is a local optimum

The notion of excursion sets suggests two percolation problems �i� At which level

E does AE cease to be a single cycle) �ii� At which level E does AE decom�

poses into many small cycles� as opposed to consisting of a single giant component

containing almost all vertices of AE and a number of very small islands) Both

problems have not been treated so far� although they seem to be of utmost im�

portance for the understanding of adaptation on combinatory landscapes In this

contribution we shall be content with investigating the structure of landscape at

�tness levels for which the cycles are still large in general

��Random Walks on Excursion Sets

Instead of performing the random walk on the entire con�guration space C one may

con�ne it to an excursion set AE � C The random walks is then automatically

constrained to a connected component of AE � ie� to a cycle We used the following

procedure to generate a walk within a cycle CE The process starts in a vertex

a� known to be in the desired excursion set These initial points are generated

by screening a large number of random con�gurations �Alternatively one might

use con�gurations obtained from some simple optimization heuristics as starting

points for higher excursion levels This would� however� bias the the sampling�

since con�gurations in large �mountains� would be favored� Then an attempt is

made to move to a neighboring vertex If it is contained in the same cycle� ie� if its

�tness is above the threshold level E then it is accepted� otherwise the attempt is

rejected The �time�series� is formed by the accepted moves only This procedure

generates a time series provided CE contains more than one con�guration In fact�

we are only interested in large cycles

� ��


It is clear that con�ning random walks to cycles means that they sample predom�

inantly in the vicinity of local optima One can hope� therefore� that the resulting

time�series provide information about the most interesting regions of the �tness

landscape ' the region of high �tness The major drawback is that� by equ��

the time series contains a superposition of two e�ects� namely the correlation of

�tness values on the landscape and the geometrical relaxation of the walk in CE

The correlation of a walk in CE is

rE�s� !Xd

�Esd�E�d� �E�hd�s�iE �� with hd�s�iE !

Xd

�Esdd� ��

Here �E�d� is the correlation of the restriction of the landscape to the excursion

set AE and hd�s�iE describes the geometric relaxation of a random walk in a cycle

CE Since the topology of CE is not known it is very di�cult to retrieve more than

qualitative information on the structure of the mountainous parts of the landscape

� ��


�� Self�A�ne Time Series and Fractal Landscapes

�� SelfA�ne Time Series

De�nition� A time series fFtg is self�a�ne �or fractal� if

s�H �Ft�s � Ft�d

�� Ft�� Ft��

where s is the number of steps between the two measurements The notation d

��

indicates equality in the sense of distributions The parameterH ful�ls � � H � �

An example is fractional Brownian motion� see� eg� �Mandelbrot� �� The

power spectrum of a time series with a distribution ful�lling �� follows a power

law �Mandelbrot � vanNess� �� of the form

S�� ! ��a with a ! � $ H ��

In case of fractional Brownian motion in continuous time� the parameter H and

the Haussdorf dimension DH of the resulting curve are related by DH ! H $ �

We remark that a time series ful�lling �� strictly for all s cannot be stationary

Instead of using the power spectrum one can use more direct methods for char�

acterizing a self�a�ne time series Probably the most immediate approach is to

consider the jump size

J�s� ! hjFt�s � Ftji ��

As an immediate consequence of �� we have J�s� � sH H can be obtained by

means of a least square �t from a log�log plot� see� eg� �Osborne � Provenzale�

�� A closely related technique has been proposed by Sorkin �Sorkin� ��

Multiplying �� by itself and taking the expectation yields

s��H h �Ft�s � Ft�� i ! h �Ft�� Ft�

� i ��

� � �


and one obtains the slope H from a log�log plot of the mean square di�erences

versus the lag s

Another approach to self�similarity focuses on the curve length as function of the

yard�stick length used for the measurement The method outlined below was

proposed in ref �Higuchi� �� as an improvement of the procedure given by

Burlaga and Klein �Burlaga � Klein� �� It provides numerically stable scaling

exponents even for a small number of data points We divide the time series fFtg

into k partial series

Fm�s� ! fFm� Fm�s� Fm��s� � � � � Fm�bN�m

scsg�

and de�ne the length of Fm�s� as

Lm�s� !N � �

sbN�ms c

Xi

jFm�is � Fm��i��sj�

The curve length L�s� measured with step size s is then the average value taken

over all the partial series

L�s� !�

s

sXm��

Lm�s��

If the time�series is self�a�ne� then the curve length follows a power law of the

form L�s� � s�D The correction factor in the de�nition of Lm�s� approaches �

for large data sets� and hence we �nd as an immediate consequence of equ��

that

L�s��N

ssHh jFt�� Ftj i � sH��

and therefore D ! ��H The parameters a� H� and D of a self�a�ne time series

are related by means of the equations

a ! � $ H ! �� D� ��

Independent estimates of a� H� and D can thus be used to determine to what

extent a given time series is consistent with the assumption of self�a�nity

� ��


��Fractal Landscapes

It is obvious that a time series obtained from a random walk on a landscape cannot

be strictly self�a�ne since it must be �at least approximately� stationary Hence

�� is an approximation that holds only for s n� where n is the maximal

distance in con�guration space

Dividing Sorkin�s equ�� by twice the ��nite� variance �� of the landscape and

substituting equ�� we �nd

s��H�� r�s�� ! �� r��

Solving for the autocorrelation function yields r�s� ! � � cs�H The parameter

c can be obtained as follows Since a single step along the random walk always

leads to distance � we have r�� ! �� def

�� the nearest neighbor correlation of

the landscape Thus ! � � c� and we �nally obtain an autocorrelation function

of form

r�s� ! �� s�H � ��

Equ�� holds of course only for s small compared to the maximum distance in the

landscape It has been used for a classi�cation of rugged landscapes �Weinberger

� Stadler� �� Stadler� ��b� in terms of the parameter

H !�

ln�� r�s��

ln s��

for small s

�� AR ��Landscapes� are Locally Fractal

An AR�� or Ornstein�Uhlenbeck� process is de�ned by the following recurrence

relation� see� eg� �Papoulis� �� Feller� ��

Ft ! �Ft�� $ �t� ��

� ��


where �t is denotes Gaussian white noise with variance �� The resulting time

series is stationary and has the Markov property Its autocorrelatation function is

r�s� ! s ! exp��s�� def

��

ln ��

where � is the correlation length as de�ned in �Weinberger� �� Fontana et al��

�� Conversely� any Gaussian stationary Markov process has an autocorrelation

function of the form �� The parameter measures the correlation of the time

series If � then the time series is almost uncorrelated� ie� fFtg is almost

white noise On the other hand� for � the time series approximates Brownian

motion

Before we proceed let us brie y discuss the relation between � and (� de�ned in

section The RNA free energy landscapes and almost all of the model landscapes

that have been investigated so far have correlation length that scale linearly with

n� see eg �Schuster � Stadler� �� for a recent overview In other words� we

have def

�� x where x scales as ��n for large systems If r�s� is of the form

�� then we �nd

� !�

x��

$

�

�x$O�x��

(� !�

x��

��

Thus � and (� di�er only by a contribution of order �� n for the landscapes

of interest

The power spectrum of an AR�� time series is

S�� !��

�� $ � � cos��

see� eg� �Yaglom� �� In fact� the Wiener�Khinchin theorem� equ�� shows

that equ�� hold for all elementary landscapes� irrespective of the distribution

function of the �tness values

Weinberger �Weinberger� �� called a landscape f C � IR an AR�� landscape

if the time series obtained by a random walk on the landscape is Gaussian and

has an autocorrelation function of the form �� The parameter describes

� ��


the ruggedness of the landscape The landscapes with exponential autocorrelation

functions are exactly the elementary landscape discussed above An AR�� land�

scape in the sense of Weinberger is thus an elementary landscape with a Gaussian

�tness distribution A number of model landscapes have been shown to be elemen�

tary �Grover� �� Weinberger � Stadler� �� Stadler� ��c� Most of them

have in fact a Gaussian distribution of �tness values� at least asymptotically as a

consequence of the central limit theorem The best known examples are the p�spin

models� the graph bipartitioning problem� graph matching� graph coloring� and

symmetric traveling salesman problems Kau�man�s Nk models are approximately

AR�� their decomposition into elementary components is discussed in detail in

�Stadler � Happel� �� The class of landscapes that are approximately AR��

includes a variety of landscapes based on RNA secondary structures �Fontana

et al�� Fontana et al�� Bonhoe�er et al�� Tacker et al��

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0rho

0.0

0.1

0.2

0.3

0.4

0.5

H, 1

-D

Figure �� Approximated scaling exponents H solid line� and ��D dotted line� as a functionof the correlation � of an AR� process� The deviations are of the order of ��

The following considerations� like equ�� depend only on the form of correla�

tion function r�s�� not on the distribution function of the �tness values They are

� ��


therefore valid for any elementary landscape The linear approximation

r�s� �� s�� s n ��

of equ�� is a good approximation for highly correlated landscapes� ie� for

landscapes with correlation lengths � ! O�n� By comparing equ�� with

equ�� we observe that elementary landscapes with large correlation length are

locally self�a�ne� with scaling parameter H!�� ie� time series obtained from

such landscapes behave locally like ordinary Brownian motion

Surprisingly� however� we �nd that even AR�� time series with small correlation

length show approximate power laws for J�s�� equ�� and for the curve lengths

L�s� Numerical simulations show that we have H � � for � �� while � � �

yields H � �� Data obtained from direct measurement of H� according to

equ�� and estimates of the scaling exponent D of the curve length obtained

from �� are consistent with each other Best �ts of the characteristic exponents

H and � �D as functions of are shown in Figure � It is interesting to note in

this context that certain log�normal distributions can also mimic ��f spectra in a

limited frequency domain �Montroll � Shlesinger� ��

� ��


��RNA Free Energy Landscapes

Folding biopolymer sequences into structures is a central problem in molecular

biology research Both robustness and accessibility of structures� as functions of

mutational change in the underlying sequence� are crucial to natural as well as

molecular evolution applied to biotechnology RNA molecules are an excellent

model system In fact� they are the only class of biopolymers for which the folding

problem has been solved at least at the level of secondary structures

An RNA sequence is a string of length n composed of an alphabet of size � In

nature the alphabet consists of the � ! � bases Guanine� Cytosine� Adenosine�

and Uracile In this paper we shall also consider the restricted alphabet fG�Cg

with � ! A natural distance between sequences is the Hamming distance

measuring the number of positions in which two sequences di�er �Hamming� ��

The con�guration space is hence a generalization of the Boolean hypercube known

as the sequence space

A secondary structure is tantamount to a list of Watson�Crick type and GU base

pairs Such a structure can be uniquely decomposed into structural elements that

are �i� base pair stacks� �ii� loops di�ering in size �number of unpaired bases�

and branching degree hairpin loops �degree one�� internal loops �degree two or

more�� and �iii� bases which are not part of a stack or a loop are termed external

�freely rotating joints and unpaired ends� Each stack or loop element contributes

additively to the overall free energy of the structure These energy terms are

empirically determined parameters that depend on the nucleotide sequence �Freier

et al�� The folding process considered here maps an RNA sequence into a

secondary structure minimizing free energy This structure can be computed using

a dynamic programming algorithm �Zuker � Stiegler� �� Zuker � Sanko��

�� The implementation used in this contribution is described in detail in

�Hofacker et al�� it is available as a public domain package �Hofacker et al��

��

In this contribution we focus not on the secondary structures themselves but rather

on the free energies� &G� of structure formation The bulk properties of these

� ��


-1.5 -1.0 -0.5 0.0 0.5log(omega)

0.0

1.0

2.0

3.0

4.0

5.0

log(

S(o

meg

a))

Figure �� Raw dat of the power spectrum obtained for a GC landscape with chain lengthn � �� at excursion level �G � �� Walk length is N � �� The solid line is the best�t to S�� a � with a � �� The dotted line is ��f�noise�

minimum free energy landscapes haven been studied extensively in the past �Bon�

hoe�er et al�� Fontana � Schuster� �� Fontana et al�� Fontana et al��

�� Fontana et al�� Fontana et al�� Schuster et al�� Schuster �

Stadler� �� They are typical representants of rugged landscapes

Figure shows a sample power spectrum obtained along a random walk as de�

scribed in section The data are rather noisy In order to smooth them we break

the walk into pieces of �� steps� calculate the power spectrum for each of them�

and then we average the power spectra �� steps is about twice the diameter of

the sequence space in this case� thus signi�cantly longer walks are not meaningful

because the range of local self�a�nity is necessarily restricted to a small multiple

of the the geometrical relaxation time � of the random walk fxtg�

hd�s�i ! hd��i �� e�s��

For a free random walk on a sequence spaces we �nd

� !��

�n$O��

� ��


Table ��Power spectrum index a for time series obtained from RNA landscapes

The values of a as obtained directly from the power spectrum are compared to the

values calculated from jump exponentH and the scaling exponentD� aH ! �$H�

and aD ! �� D

n ��

&G� a aH aD a aH aD a aH aDAUGC

� ��

GC

� ��

� �G in kcal�mol�� indicates insu�cient data�Systematic errors are estimated to be of the order of �� compare Figure �

This expression will be derived in the Appendix

The data are consistent with a ��a spectrum with a not much larger than �

Numerical values are shown in Table � It turns out� however� that the data are

also consistent with the power spectrum of an AR�� process� see Figure �

The additivity of the energy contributions implies a certain degree of neutrality in

the landscape� for details see �Fontana et al�� Several structures which con�

sist of identical sets of substructures map onto the same selective values� although

their phenotypic appearances are di�erent In fact� there are very large neutral

networks on the level of secondary structures themselves �Schuster et al��

Reidys et al�� Gr�uner� �� This implies that even at fairly high excursion

� � �


-1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4log(omega)

2.0

3.0

4.0

5.0

log(

S(o

meg

a))

Figure �� The dots are spectral data for walks on a GC landscape with n�� excursion level�� averaged over � walks� The solid curve is the best �t to the AR� power spectrum�equ�� with an estimate for the parameter � � �� corresponding to a correlationlength � � �� The correlation length estimated directly from the autocorrlationfunction is about �� as shown in ref� Fontana et al�� The corresponding powerspectrum is shown as dotted line� The solid straight line is a least square �t with apower law ��a we �nd a � ��

levels �a couple of standard deviation above the mean� the excursion sets are still

large

We �nd that the scaling properties do not depend strongly on the excursion level

There is� however� a systematic trend towards smaller values of � for higher excur�

sion levels Our data indicate that mountainous regions of the landscape are not

drastically di�erent from the average Our data are biassed by the geometry of

the cycles CE� however� and hence a detailed quantitative analysis is not possible

at present

� � �


��Conclusions

The structure of the mountainous parts of RNA free landscapes was studied by

random walks con�ned to excursion sets at given energy levels

Spectral data and local scaling analysis of the series generated by simple random

walks show self�a�nity consistent with the low�frequency behavior of an AR��

time series We �nd that in general an AR�� processe appears to be approx�

imately self�a�ne on length scales smaller than a few correlation lengths The

data obtained from RNA free energy landscapes indicate that a fractal�like struc�

ture is present at length scales up to the diameter of the sequence space This

is a consequence of the fact that the correlation length of the RNA free energy

landscapes is comparable to the sequence length

Our computer experiments exhibit no signi�cant dependence of the statistical

properties of the excursion set con�ned walk on the energy level Hence� at least

qualitatively� the statistical properties of the mountains do not di�er from the

low�lands This is true at least as long as the excursion set does not break up in

very small cycles

The present study suggests that a detailed investigation of the percolation of ex�

cursion sets� of the geometry of excursion sets� and of the geometrical relaxation of

random walks con�ned to cycles will be necessary before a complete understanding

of the structure of the mountain ranges of �tness landscape is possible

Acknowledgments

The work was funded by �OAD projno Z� ��EH�� EH�Project �� SB

gratefully thanks Prof Murali Sheshadri and Prof A Nadarajan for their interest

and support PFS thanks the Inst Ciencias Nucleares and the Inst de Fisica of

the Universidad Nacional Autonoma de Mexico for their hospitality in September

�� when this paper was �nished

� �


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Appendix Relaxation of Random Walks in Sequence Spaces

For a sequence space a detailed analysis of the geometric relaxation of a simple

random walk is possible The probabilities �sd as de�ned in sect can be obtained

recursively from

�� ! �

�sd ! � for s � d

�sd ! w��d� ��s��d�� $w��d��s��d $ w��d $ ��s��d��

�A��

where coe�cients w��d�� w��d�� and w��d� are the probabilities for making a step

forwards� backwards or sidewards given one is in distance d from the origin of the

walk For sequence spaces we have �Fontana et al��

w��d� !n� d

nw��d� !

d

n

� �

� � �w��d� !

d

n

�

� � ��A��

De�ne the moments of the distribution �sd by

&m�s� ! hd�s�mi !Xd

�sddm �A��

&��s� is then the average distance after s steps Inserting the recursion �A�� into

the de�nition of &m�s� yields after considerable algebra the following recursion

for the m�th moment

&m�s� ! � $m��X��

�m

m� �

��

m� �

�$ �

�

��

�

n

�&m��s � ��

$

�n

m��X��

�m

m� �

�&m��s� ��

�A��

This recursion is of the form �&�s� ! �$A � �&�s� �� where A is lower triangular

Hence the eigenvalues �m of A are given by the diagonal elements of A

�m ! ��m�

n

�

�� A��

The m�th moment is therefore of the form

&m�s� ! &m�� ,�� ak�sk- � �A��

The slowest mode corresponds to the eigenvalue �� The corresponding relaxation

time is

�� ! ��

ln��!

��

�n$O�� A��

for large n Explicit expressions for the long time limits &m�� of the moments

are obtained as non�zero �xed points of the recursions �A��

� � �


Table of Contents

� Introduction

Landscapes �

� Rugged Landscapes �

Correlation Functions �

� Elementary Landscapes �

� Power Spectra �

� Excursion Sets �

� Random Walks on Excursion Sets ��

� Self�A�ne Time Series and Fractal Landscapes �

�� Self�A�ne Time Series �

� Fractal Landscapes ��

�� AR��Landscapes� are Locally Fractal ��

� RNA Free Energy Landscapes ��

� Conclusions

Acknowledgments

References �

Appendix Relaxation of Random Walks in Sequence Spaces �

� i �

Approximate Scaling Properties of RNA Free Energy Landscapes...Baskaran et al Scaling in RNA...

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