NP Completeness ofKauffman’s N-k Model, aTuneably Rugged FitnessLandscapeEdward D. Weinberger

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SANTA FE INSTITUTE

NP Completeness of Kau�man�s N�k Model�

a Tuneably Rugged Fitness Landscape

Edward D� Weinberger

Max�Planck�Institut f�ur biophysikalische Chemie

Postfach ����

Am Fassberg

D�� G�ottingen�Nikolausberg

Federal Republic of Germany

The concept of a ��tness landscape�� a picturesque term for a mapping of the vertices of a

�nite graph to the real numbers� has arisen in several �elds� including evolutionary theory�

The computational complexity of two� qualitatively similar versions of a particularly simple

�tness landscape are shown to di�er considerably� In one version� the question �Is the global

optimum greater than a given value V�� is shown to be answerable in polynomial time by

presenting an ecient algorithm that actually computes the optimum� The corresponding

problem for the other version of the landscape is shown to be NP complete� The NP

completeness of the latter problem leads to some speculations on why P �� NP�

�

Introduction

The notion of an adaptive �landscape representing the abstract ��tness of various kinds

of organisms in various contexts has been a �xture of evolutionary biology ever since it was

proposed by Sewell Wright ������ Although there are problems with a notion of ��tness

that is a property of an individual� independent of other individuals and the environment�

the discoveries of molecular biology have signi�cantly reinforced the power of this idea�

We now understand� for example� the role of a discrete genomic �blueprint in specifying

the chemical constitutents of enzymes and a glimmering of how sensitive the �tness of the

organism can be to variations in enzyme chemistry� so that it makes sense to identify the

speci�c sequence of nucleotide bases in the genome as the argument of the �tness function�

It has also become increasingly clear that the �design of organisms involves a host of

complex trade�o�s� implying that there must inevitably be large numbers of local optima

in such �tness landscapes� It was thus only a matter of time before the analogy between

optimization �i�e� selection� on landscapes and combinatorial optimization problems ap�

peared in the biological literature� One paper that discussed this analogy �Kau�man �

Weinberger� ������ also proposed a simple statistical model of a �tness landscape� the N�k

model� that could be used as an aide to the qualitiative understanding of more complex� if

more realistic models� The purpose of the present paper is to consider the computational

complexity of optimization on landscapes generated with this simple model� We show� in

particular� that the optimization problem for one version of the model can be solved in

polynomial time� but that another� qualitatively very similar version of the model� is NP

complete� This result makes the N�k model an ideal candidate for investigating the nature

of NP completeness�

One might think that optimization onN�k landscapes isNP complete because a knowledge

�

of the �tness of a given con�guration says little about the �tness of distant con�gurations�

thus leading to the need for an exhaustive search over a large number of con�gurations�

Part of this conjecture turns out to be true� we can identify a set of pairwise distant

con�gurations whose number grows faster than any polynomial in N � However� when

we compute the correlation in �tness between pairs of sites in this set� the correlation is

actually larger for the NP complete version of the model than for the polynomial time

version� thus providing strong evidence that the rest of the conjecture must be false� We

conclude the paper by noting di�erences between the two models in �tree width � Dress

������ has shown that the global optimum in models where this quantity is O��� can be

solved in polynomial time� Our result � that the tree width of anNP complete problem is

O�N� � suggests that the notions of NP completeness and tree width are more generally

related�

De�nition of the N�k Model

In nature� the argument of the �tness function is the set of all possible sequences of

the four nucleotide bases� Nature �evaluates this function by translating these nucleic

sequences into sequences of amino acids� It is only amino acid sequences � the enzymes

and structural proteins of an organism � that have biological signi�cance� Therefore� the

simplest version of the N�k model ignores genetics and assigns a �tness to the sequences of

amino acids directly� The N�k model makes two further simpli�cations� the �rst� that the

amino acid sequence has a �xed length of N sites� and the second� that there are only two

possible amino acids� rather than the full complement of twenty� that can occupy a given

site of the sequence� This last assumption is justi�ed by the fact that almost all of the

properties of such sequences are determined by their three dimensional� folded structures�

which� in turn� are determined by the chemical properties of the constituent amino acids�

The most important of these properties is polarity� those amino acids in the sequence

that are polar get pulled to the outside of the folded structure by chemical attraction to

surrounding water molecules� and non�polar amino acids get pushed into the interior of

the structure�

The assumption of two amino acids per site also dramatically simpli�es the modelling task

by reducing the argument of the �tness function to a bit string� which we denote b� The

N�k model assigns a real valued ��tness to b by �rst assigning a real valued ��tness

contribution � fi � to the ith bit� bi� in b� Each such assignment depends� not just on i

and the value of bi� but also on � k � N other bits� which we call its �neighbors � The

�tness contribution of each site is a random function� fi�si�� of the substring� si� formed

by the ith bit and its k neighbors� fi�si� is assigned by selecting an independent random

variable from some distribution p�x�� such as the uniform or Gaussian distributions� for

each of the �k�� possible values of si� thus generating a ��tness table for the ith site�

There is a di�erent� independently generated table for each of the N sites� Then� given

any string of N bits� the total �tness of the string� F � is de�ned as the average of the

�tness contributions of each site� that is�

F �b� ��

N

NXi��

fi�si��

The use of a probability distribution in assigning the �tness contributions can be inter�

preted either as an admission of ignorance of the true nature of the complex couplings

between the bits or as an attempt to capture the typical statistical properties of a wide

class of landscapes with k interconnections per bit�

One other aspect of the N�k model must be speci�ed� namely� the way in which the

substrings� si� are chosen� The simplest � but not the only � way of choosing neighbors�

at least for even k� is to use the k sites adjacent to site i� that is� the bits at sites i � k��

�

thru i� k��� As in the original formulation of the model� we introduce periodic boundary

conditions to assign neighbors to sites i with i � k�� and i � N � k��� In other words�

we assume that the sites are arranged in a circle� such that site N is next to site �� Under

this assumption� if k � �� site N has neighbors N � � and �� site � has neighbors N and ��

and� more generally� site i has neighbors �N � i � k��� mod N� � � � � �N � i� k��� mod N

for any even integer k� This assignment of the neighbors gives rise to a class of short range

spin glasses� Alternatively� we could assign the neighbors by randomly selecting� for each

site i� k other sites on the string to be used in forming the index to the ith �tness table�

a de�nition that also makes sense for odd k� This assignment of neighbors makes the

model similar to a long range� dilute spin glass� Rather surprisingly� local features of the

landscape � the height of local optima� and the length of typical �up hill walks through a

series of �tter one mutant variants to these local optima � were remarkably insensitive to

the details of how the k�� bit substrings were chosen in computer simulations �Kau�man�

Weinberger� � Perelson� ����� Kau�man � Weinberger� ������

The N�k model a�ords a �tuneably rugged �tness landscape� since tuning k alters the

ruggedness of the landscape� For k � � each site is independent of all other sites� Either

the bit value or the bit value � is almost surely ��tter than the other� hence� a single

speci�c sequence comprised of the �tter bit value in each position is almost surely the

single� global optimum in the �tness landscape� Any other string is sub�optimal� and lies

on a connected walk via ��mutant �tter variants to the global optimum by �ipping bits

from less �t to more �t values� The length of the walk is just the Hamming distance from

the initial string to the global optimum� For a randomly chosen initial string� half of the

bits will be in their less �t state� hence the expected walk length is just N��� A transition

to a one mutant neighbor �i�e� the �ip of a single bit� typically alters �tness by an amount

O���N�� In contrast� the fully connected N�k model yields a completely random �tness

�

landscape� For this k � N � � case� the �tness contribution of each site depends on all of

the other sites because the �context of each of the N � � other bits is changed when even

a single bit is �ipped� In this case� therefore� the �tness of each N bit string is statistically

independent of its neighbors� As was shown in Kau�man � Levin ������� Weinberger

������� Macken � Perelson ������ and Weinberger �����a�� such random landscapes have

very many local optima ��N��N ���� on average�� walks to optima are short �O�lnN�� on

average�� and only a small fraction of local optima are accessible from any initial string�

Thus adaptive walks vary dramatically as the ruggedness of the landscape varies�

The N�k decision problem

Having de�ned the model� we are now in a position to consider the following decision

problem� Is the global optimum� FMAX� in a given instance of an N�k landscape greater

than some speci�ed value V� In view of the work of Dress ������� it is not surprising

that this question can be answered for adjacent neighborhoods with periodic boundary

conditions by the following simple dynamic programming algorithm that actually �nds the

globally maximal �tness� For simplicity� we present the algorithm for k � �� leaving the

trivial generalization to arbitrary k to the reader� Also� N is to be added or subtracted� as

appropriate� to subscripts outside the range �� �� � � � �N so that they assume values within

that range�

Let fbi��bibi��i be the site �tness of the ith site� given the values of the bits bi��� bi� and

bi��� and let FbNb�jbi��bi��i be the maximum value of the sum

i��Xj��

fbj��bjbj��j �

over the values of b�� b�� � � � � bi� given the values of the bits bN � b�� bi���and bi��� The

algorithm then consists of the three phases

�

Initialization� FbNb�jb�b�� � fbNb�b�� � fb�b�b�� �

Continuation� FbNb�jbi��bi��i � max

bi

�FbNb�jbibi��i�� � f

bibi��bi��i��

�for � � i �

N � �� which implicitly speci�es a value for bi for � � i � N � � and each

choice of bN � b�� bi��� and bi��� Use the value of bi thus speci�ed for subsequent

calculations�

Note that the formula still makes sense when i � N � � or i � N � �� but there

will be only � FbNb�jbN��bNN�� values and only � F

bNb�jbNb�N�� values instead of the ��

FbNb�jbi��bi��i values de�ned for � � i � N � ��

Termination� FMAX � maxbNb�

�FbNb�jbNb�N��

��

We have therefore proven

Theorem � The N�k decision problem with adjacent neighborhoods is solveable in

O��kN� steps� and is thus in P�

However� the situation for random neighborhoods is quite di�erent� as is shown by

Theorem The N�k decision problem with random neighborhoods is NP complete for

k � �

Proof We note �rst that the single integer N characterizes the size of the problem com�

pletely if k is �xed� �k��N real numbers are required to specify the �tness tables� and k

integers per site� or kN integers per instance of the problem to specify the neighborhoods�

Furthermore� it is merely a matter of a table lookup followed by an addition for each site�

or O�N� total work to check that a proposed solution does� indeed� have a �tness greater

than a given value� We conclude that the N�k decision problem is in NP�

�

We demonstrateNP completeness by showing that theN�k problem is polynomially equiv�

alent to one of the best known NP complete problems� the SAT problem �Garey � John�

son� ������ Given N boolean variables b � �b�� b�� � � � � bN � and a list of M expressions�

Ei�bpi � bqi � bri �� involving arbitrary triples of these variables and the operators AND� OR�

and NOT� the SAT problem is to determine whether values exist for the bi�s such that

all of the expressions are satis�able �i�e� all evaluate to TRUE�� If we can show that every

such problem can be mapped into an N�k decision problem in polynomial time� we can

conclude that the latter problem is �at least as hard as the SAT problem� because our

mapping� together with a polynomial time solution to the N�k decision problem� would

provide a polynomial time solution to the SAT problem�

It is easiest to generate the N�k decision problem that corresponds to a given SAT problem

when N � M and k � � We �rst transform each expression Ei�bpi � bqi � bri � into the

equivalent expression

E�

i�bi� bpi � bqi � bri � � Ei�bpi � bqi � bri � AND �bi � bi��

where the expression a � b is TRUE if and only if a � b� We then identify bi with the ith

bit in the N�k con�guration� and the other variables appearing in the ith expression with

bi�s �neighbors � We assign the �tness tables associated with bi as follows� a �tness table

entry is assigned the value � if the corresponding E�

is TRUE for the speci�ed values of

the b�s� and if it is FALSE� Clearly� the corresponding SAT problem is solved if we can

determine whether the global maximum of the N�k landscape thus generated is greater

than or equal to N �

A trivial variation of the above mapping su�ces whenM � N � As before� we identify each

SAT variable� bi� with the ith bit in the N�k con�guration� modify the M expressions

such that bi appears in the ith expression for � � i � M and assign the �tness tables as

�

described above� SitesM ���M��� � � � �N are assigned �tness tables in which every entry

is given the value �� Once more� the given SAT problem is solved if and only if the global

maximum of the corresponding N�k problem is at least N �

TheMN case can be handled by introducing additional variables bN��� bN��� � � � � bM � and

proceeding as above� provided we lengthen the N bit string in the N�k model to M bits�

and assign the neighborhoods as before�

The N�k decision problem for k is a fortiori NP complete because every SAT problem

can be embedded in a k decision problem by de�ning the expressions

E��

i �bi� � � � � bj � bpi � bqi � bri � � Ei�bpi � bqi � bri � AND �bi � bi� AND � � � AND �bj � bj ��

where the bits bi� � � � � bj include the bits at the ith site and some arbitrary collection of

k� neighbors� The previously given mapping to the N�k problem can then be applied�

Remark �� So far as we know� the question of whether the k � � random model is NP

complete is open�

Remark � An almost identical argument shows that the �P �T model� proposed by

Pedro Tarazona� in which the �tness table entries for site i do not depend on the ith bit�

but only on its k neighbors� is also NP complete for k � � The computational complexity

of the k � � P �T model is also an open question� but we conjecture that it is polynomially

equivalent to the polynomial time �SAT problem� a variant of the SAT problem in which

the M boolean expressions involve arbitrary pairs� rather than triples� of the boolean

variables�

NP Completeness and Correlation

The foregoing is a clear example of Garey and Johnson�s observation that seemingly trivial

�

modi�cations to a combinatorial optimization problem can render it intractable� Previous

analytical and numerical work suggests that many of the statistical properties of N�k

landscapes for large� but �xed k are quite similar for adjacent and random neighborhoods�

Weinberger �����a� shows that the mean number of local optima� distances between optima

and the expected �tness of a local optimum is� asymptotically for large k� the same in

both cases� In Appendix I� we compute the exact correlation R�d�� between pairs of points

separated by a Hamming distance d� from which we can deduce the limiting behavior for

d� k �� N � For the random landscape� we obtain

R�d� � ��d�k � ��

N�d�d� ��k�k � ��

�N��O

��dkN

����

for the adjacent neighbor landscape� we obtain the rather similar expression

R�d� � ��d�k � ��

N�d�d� ��k�k � ��

�N��O

��dkN

����

These correlations� along with the common mean and variance of the �tnesses� completely

characterize the multivariate Gaussian distribution to which the distribution of �tnesses

converges as N� k ���

We might conjecture that a problem is intractable because the correlation between pairs

of distant con�gurations decays so rapidly as their distance increases that their �tnesses

are e�ectively independent� We then have the obvious� but suggestive

Lemma A Every algorithm for �nding the maximum �or minimum� of M completely

arbitrary real numbers requires at least O�M� steps�

Proof If such an algorithm required merely o�M� steps� some of the numbers must neces�

sarily be ignored by the algorithm� The answer given must then be independent of which

�if any� of the ignored numbers is the maximum� Clearly� such an algorithm cannot be

guaranteed to �nd the maximum in all cases�

�

Any hope of �nding the maximum of the �N �tnesses assigned to the �N vertices in

polynomial time must be based on the ability to rule out whole classes of �tnesses with a

single operation� which� from the above lemma� is impossible if the vertices are assigned

completely independently and without any a priori knowledge about the assignment� On

the face of things� it would seem that such problems cannot possibly lie in NP� however�

we show that a similar problem is embedded in the problem of optimization on a random

neighbor N�k landscape� for which the correlation between �tnesses of points separated

by a Hamming distance of N�� is ��e�k��

�� �O�k�N�

��See Appendix I��� Clearly� this

quantity can be made as small as desired by choosing k su�ciently large and letting N tend

to in�nity �In fact� the N�k decision problem remains in NP for k � O�logN���� Because

the �tnesses are asymptotically jointly Gaussian� an asymptotically vanishing correlation

between them implies that they are asymptotically statistically independent� and therefore

a knowledge of one �tness tells us� asymptotically� nothing about the other� We now prove

Lemma B For arbitrarily large N � there exists a set� N � of �N �log�N���� bit strings

of length N such that the Hamming distance between any pair of strings is at least N���

The proof of Lemma B is based on

Lemma C For arbitrarily large N � there exists a set� !N � of �N bit strings of length N

such that the Hamming distance between any pair of strings is at least N���

Proof of Lemma CWe start by recursively constructing� for N � �n� n � �� a set� !N � of

�N strings of length N whose pairwise Hamming distances are at least N��� Clearly�

!� � f� ����� ��� ��� ��� ��� ��� ��g�

The �� strings of length N � � in ! are obtained as follows� form � complementary pairs�

�ti� "ti�� from the strings ti !�� Because the distance between distinct elements of !� is

��

at least �� the requisite N � � strings are tijjti� tijj"ti� "tijjti� and "tijj"ti� where the symbol

�jj denotes concatenation �simple juxtaposition�� The pairwise distance between each of

the �� strings thus formed is least N�� � �� either because the strings are formed from

di�erent t�s� or because half of one string is the complement of the corresponding half of

the other string� The resulting set is also a set of � pairs of complementary strings� so that

the iteration can be repeated once again� and� in fact� arbitrarily often� each time doubling

the number of strings� The resulting set of �N strings is obviously not unique� because

another set can be generated by complementing the bits at a �xed position in each of the

strings in the �rst set�

Remark� In the course of referreeing this paper� Prof� Andreas Dress proved my conjec�

ture that !N has maximal size� in that at least one member of !N is fewer than N�� bit

�ips away from all N bit strings not in !N � His proof begins by identifying the successive

bits bi� for � � i � N of the string S as the image of a mapping from the index� i� to the

appropriate bit value� He then notes the equivalence of the integer i � � with its binary

representation� the n � log�N bit string� which we write explictly as ���� � � � �n� The

construction above shows that !N consists of precisely those N bit strings S whose ith

bit� bi is given by

�� �

nXj��

�j�j �

for some choice of the bits ��� ��� � � � � �n� and for all i between � and N � The strings thus

generated are precisely the set of a�ne mappings from the n bit strings ���� � � � �n to single

bits� We can thus re�establish that the strings in !N have pairwise Hamming distances of

at least N��� If the corresponding ��s for two mappings di�er at position p� the strings

they produce will di�er at indicies i whose binary representations have the property that

X�p � ��

and there are exactly N�� such indicies� If the ��s di�er only at position p � � the

��

corresponding strings will be complementary�

The maximality of !N now follows from character theory� Identify !N with the irreducible

characters of Fn� � considered as a group� via the mapping

�i � ����bi � ����

���P

n

j���j�j �

where �i is the ith component of the vector ��� Fn� � For a given N bit string S� denote

this mapping as ����S�� Now consider the inner product

� ����S�� ����S�

� ��

�n

Xi

�i��

i

��

�n

hd�S�S

�

� � d�S� "S�

�i �

where d�S�S�

� is the Hamming distance between S and S�

� The fact that !N is closed

under complementation guarantees that the Hamming distance between it and any N bit

string S is at most N��� If S could be exactly N�� bit �ips from every member of !N �

the di�erence d�S�S�

�� d�S� "S�

� would be exactly zero for all S�

!N � and thus the inner

product of ����S� with all of the irreducible characters of Fn� would be zero� As is well

known� this last condition is satis�ed if and only if every component of ����S� is zero� which

is clearly absurd�

We now return to the

Proof of Lemma B Clearly� the �� strings of length � whose pairwise Hamming distances

are at least � are the binary representations of the integers thru ��� For larger values of

N � we form the strings of N from two N�� bit substrings� a pre�x � and a su�x � �

is an arbitrary member of N��� is constructed from an arbitrary member !N�� by

complementing those bit positions marked by ��s in � �i�e� � � �� where the addition

is taken modulo ����

To check that this construction does� indeed� meet the requirements stated above� we

partition N into subsets of strings with the same pre�x� Clearly� there is no problem with

�

the strings in the same subset� because their su�xes� corresponding to di�erent members

of !N��� are mutually separated by at least N�� bits� There is also no problem with

strings in di�erent subsets whose su�xes were derived from the same � Their pre�xes�

� and ��

must di�er in at least N�� bit positions� so that their su�xes� � � � and

�

� ���

� must also di�er in at least N�� bit positions� There remains the case in which

two strings in N have both di�erent pre�xes and di�erent su�xes� Because they have

di�erent pre�xes� the above construction guarantees that their pre�xes di�er in at least

N�� positions� we now show that the same applies to the su�xes� and �

� We have

� d�� �

� � d�� � � ��

� �

��

For each N � !N N � so that� in particular� � �

N��� It follows from the fact �proven

below� that N�� is closed under �� and the positive distance between �� and ��

� �

that they are distinct members of N��� and therefore di�er in at least N�� bit positions�

We now verify that N is closed under �� by induction� For N � �� closure obtains

trivially� For larger N � we assume closure for N��� and establish it for N � Given two

elements of S�� S� N � which we write ���jj����� and ���jj������ where ��� �� N���

�� � !N��� we have

S� � S� � ��� � ��� jj #��� � �� � ��� � ��$�

Using the induction hypothesis� �� � �� N��� using the commutativity and the asso�

ciativity of �� � we write the su�x string as ��� � ��� � �� � ��� We now observe that

!N�� is also closed under �� � as is clear from their construction� all elements !N��

are N�� bit strings formed by choosing a single element t !� and forming arbitrary

concatenations of t and its complement� "t� Thus� � � � is the arbitrary concatenation of

the four bit strings t� � t� � "t� � "t� and t� � "t� � "t� � t�� From the fact that these last

��

two strings form a complementary pair in !�� we conclude that � � � is also in !N���

allowing us to write

S� � S� � ��jj�� � �

for some �� N�� and some � !N��� as required�

We now count the number of distinct strings� DN � in N � Because each string in N is

generated by choosing one of the DN�� members of N�� and one of the N members of

!N��� we have the recursion relation

DN � NDN���

Writing N � �n� and Gn � D�n � this relation becomes

Gn � �nGn���

Given the initial condition D� � G� � �� that was computed �by hand above� we see

that Gn � �n�n������� so that DN � �N �log�N����� as claimed�

Remark� We conjecture that N is also maximal�

We state the conclusions of this section in the following

Summary The NP complete random neighbor N�k optimization problem seems to be

at least as hard as the problem of optimization over a sample of DN � �N �log�N����

Gaussian random variables whose pairwise correlations can be made as small as desired�

so that they are �arbitrarily close to being pairwise statistically independent� Because

DN grows faster than any polynomial in N � this result seems to suggest that P �� NP�

Unfortunately� small correlations between �tnesses of distant points on the landscape are

not� in themselves� su�cient to render a problem intractable� However small the correla�

tions between the �tnesses of strings in N can be made in the random neighbor landscape�

��

they can be made even smaller in the adjacent neighbor landscape� at least for su�ciently

large landscapes� In fact� if we ignore the O���N� correction terms in the correlation

functions� we have the relation

Radj��� � ��� ��k�� � Rrand��� � �� � ��e��k�

which follows immediately from the inequality �� � � e�� � for � � � ��

Discussion

The crucial di�erence between the adjacent and random landscapes seems to be the number

of bits upon which each site �tness could depend� the �tree width discussed by Dress

������� For the adjacent neighbor case� we know a priori that each site depends only on

the bit at that site and the k bits at adjacent sites� so that the tree width is k � �� It

is this a priori knowledge that makes possible the solution of the corresponding decision

problem with a polynomial time dynamic programming algorithm� In contrast� which k��

of the N bits each site �tness depends will vary from one instance of the random neighbor

model to the other� so that the tree width is no longer k � �� but O�N�� It appears�

therefore� that the random neighbor problem is intractable� not only because of the small

correlations between distant points on the landscape� but also because there is no e�ective

way to partition the problem into smaller problems and thus use information available

from nearby points to infer relationships between more distant points�

In Weinberger� ������ Weinberger �����a� and Weinberger �����b�� we have argued for

the importance of the N�k model and� more generally� the notion of statistically isotropic

�AR��� landscapes � In general� statistical isotropy implies that the correlation� R�d��

between the �tnesses of pairs of points depends only on the distance d between them�

in the speci�c case of the AR��� landscape� this correlation function assumes the form

��

R�d� � e�d�T � for some choice of the �correlation length � T � The exact calculations of

the pair correlations for the random neighbor and adjacent neighborN�k models presented

in the Appendix show some disparity from the precise �AR����ness of the P �T model�

However� this disparity is relatively minor when d � T � and it is R�d� values for d � T that

determine the local properties of the landscape �See the above cited papers for details���

From these observations� we make two conjectures� that� in general� approximately AR���

landscapes with unbounded tree width are NP�complete� but these problems have statis�

tical properties similar to the properties of �easy optimization problems� In view of the

ubiquity of AR��� landscapes in optimization problems in computer design �Sorkin� �����

and in RNA folding landscapes �Fontana et al� ������� this conjecture deserves further

study�

Finally� our results suggest that nature is solving a non�trivial optimization problem in

the �design of individual enzymes� at least if the relevant �tness landscape resembles

an N�k or P �T landscape� As we saw above� the �tness of an enzyme depends crucially

on its three dimensional structure� so that the random neighbor N�k landscape �i�e� the

NP�complete one� is clearly more biologically accurate than the adjacent neighbor model�

This observation is further evidence that there is much to be learned about optimization

from the study of evolutionary strategies�

Acknowledgements

The author gratefully acknowledges the support of ONR Grant N������K���� for the

time period in which this work was begun and the support of a Max Planck Stipendium

during its gestation� A discussion with Andreas Dress convinced the author that the opti�

mization problem on adjacent landscapes could be solved in polynomial time via dynamic

programming� and his subsequent encouragement resulted in the completion of this work�

��

including the correction of some mistakes in a previous version and the proof that !N

is maximal� The author would also like to acknowledge useful discussions with Stuart

Kau�man on the general subject of evolution as a combinatorial optimization problem�

and Pedro Tarazona for suggesting the P �T model and for the computation of both its

correlation function and the correlation function of the random neighbor N�k model�

References

Dress� A� ������ �On the Computational Complexity of Composite Systems� Lecture Notes

in Physics� Vol� �� � Fluctuations and Stochastic Phenomena in Condensed Matter� L�

Garrido �ed��� Springer� Berlin�

Fontana� W�� Griesmacher� T�� Schnabl� W�� Stadler� P�� and Schuster� P� ������� �Statis�

tics of Landscapes Based on Free Energies� Replication and Degradation Rate Constants

of RNA Secondary Structures� Monatshefte f�ur Chemie� in press�

Garey� M� and Johnson� D� ������� Computers and Intractability A Guide to the Theory

of Incomputability� W� H� Freeman� San Francisco�

Kau�man� S� � Levin� S� ������� �Towards a General Theory of Adaptive Walks on

Rugged Landscapes� Journal of Theoretical Biology �� ������

Kau�man� S�� Weinberger� E�� and Perelson� A� ������� �Maturation of the Immune

Response Via Adaptive Walks On A�nity Landscapes� Theoretical Immunology� Part I�

Santa Fe Institute Studies in the Sciences of Complexity� A� S� Perelson �ed��� Addison�

Wesley� Reading� Ma�

��

Kau�man� S� �Weinberger� E� ������� �TheN�kModel of Rugged Fitness Landscapes and

Its Application to Maturation of the Immune Response� Journal of Theoretical Biology

���� No� �� ����

Macken� C� and Perelson� A� ������� Protein Evolution on Rugged Landscapes� Pro�

ceedings of the National Academy of Sciences � �����

Sorkin� G� ������� �Combinatorial Optimization� Simulated Annealing� and Fractals �

IBM Research Report RC���� �No� ������

Weinberger� E� ������� �A More Rigorous Derivation of Some Results on Rugged Fitness

Landscapes� J� theor� Biol� ��� No� �� ��������

Weinberger� E� ������ �Correlated and Uncorrelated Fitness Landscapes and How to Tell

the Di�erence� Biological Cybernetics �� No� �� �����

Weinberger� E� �����a�� �Local Properties of Kau�man�s N�k model� a Tuneably Rugged

Energy Landscape� Physical Review A� ��� No� �� ��������

Weinberger� E� �����b�� �Fourier and Taylor Series on Fitness Landscapes� Biological

Cybernetics� �� ����

Wright� S� ������ �The roles of mutation� inbreeding� crossbreeding and selection in evo�

lution� In� Proceedings �th Congress on Genetics �����

��

Appendix

Calculation of the Correlation between Pairs of Fitnesses in the N�k Model

We want to compute

R�d� �E#f�a�f�b�$ � ��

��

where a and b are bit strings separated by Hamming distance d� f�a� and f�b� are their

respective �tnesses� � is the common mean of these �tnesses� and �� is their common

variance� The expectation� E� is taken over the joint distribution of the random variables

f�a� and f�b�� Without loss of generality� we choose a distribution for the �site �tnesses

that has mean zero and variance �� We then have � � � �� � ��N � and

R�d� ��

NE

��� NX

j��

fj �a�

A�Xj ��C

fj �a� �Xj�C

fj�b�

A�� �

where the notation is chosen to re�ect the fact that a certain subset� C� of the site �tnesses

change when bit string a is changed to bit string b� but site �tnesses fj � C� i�e� all of the

others� remain the same� For pairs of sites i �� j� the de�nition of the N�k model guarantees

that site �tnesses fi�a� and fj�b� are independent� and thus uncorrelated� Similarly� fj�a�

and fj�b� are either identical �because bit j and its neighbors are identical in both a and

b� or they are independent� In any case� we conclude that

R�d� ��

NEhXj ��C

f�j �a�i

� Prfj � Cg�

In other words� the correlation between the �tnesses of two bit strings in the N�k model is

exactly the probability that a randomly chosen site �tness is the same in the computation

of f�a� and f�b��

For the random neighbor model� the required probability is easily obtained� A site �tness

is unchanged only if the bit at that site is not one of the d bits that has been �ipped and if

�

it is not one of the k neighbors of any �ipped bit� The probability that a site satis�es the

�rst condition is �� d�N � the probability that a site satis�es the statistically independent

second condition is #�� k��N � ��$d� Thus� for the random neighbor model�

R�d� �

���

d

N

����

k

N � �

�d

�

For k� d �� N � we have

R�d� � ��d�k � ��

N��d� ��dk�k � ��

�N��O

��dkN

����

If d � �N � where � � O���� and k �� N �

R��N� � ��� ��

���

k

N � �

��N� ��� ��e��k

�� �O�k�N�

��

For the P �T model� in which neighborhoods are de�ned completely at random� the proba�

bility that a site is not a�ected by �ipping a random bit is ��k�N � independent of whether

other bits have been �ipped previously� provided they have not a�ected the given site� The

probability that a site remains una�ected by d such �ips is thus ��� k�N�d� For purposes

of comparison with the other models� we give the small and large d approximations to R�d�

for the P �T model with k � �� rather than k neighbors� The �rst of these approximations

is

R�d� � ��d�k � ��

N�d�d� ���k � ���

�N��O

��dkN

���for d� k �� N�

as is clear from a binomial expansion of � � �k � ���N

�d� If d � �N � where � � O����

and k �� N �

R�d� � e���k���� �O

�k�N

���

The same derivation for the adjacent neighbor model begins by imagining that the N site

�tnesses are arranged in a circle� and that the �ipped bits are represented by the integer

��

vector �n�� n�� � � � � ni� � � � � nd�� where � � ni � N � Without loss of generality� we choose

n� � � and n� � n� � � � � � nd � N � Because we must make d � � choices among the

N�� remaining bits� and� because we cannot make the same choice twice� there are N��d��

�ways to make the required choices� However� if we constrain ni�� � ni � l� we need only

make d� � choices from N � l � � bits�

The number of ways of making this second set of choices is N�l��d��

�� The probability� �l�

that ni�� � ni � l is then given by

�l �

N�l��d��

� N��d��

� �

independent of i� If l � k� then l site �tnesses change as a result of �ipping bit ni���

otherwise� k � � site �tnesses change� It follows that the expected number� E#jCj$� of

changed site �tnesses after moving a distance d from the starting point is

E#jCj$ � d

�kXl��

l�l � �k � ��h��

kXl��

�l

i�

� d�k � ���d

N��d��

� kXl��

�k � �� l�

�N � l � �

d� �

��

The probability that a randomly chosen site �tness doesn�t change is then � � E#jCj$�N �

and

R�d� � ��E#jCj$

N

� ��d�k � ��

N�

d

N N��d��

� kXl��

�k � �� l�

�N � l � �

d� �

�

For d� k �� N � this expression can be written as

R�d� � ��d�k � ��

N�d�d � ��

N�

kXl��

�k � �� l�

��� l��

N

���� l��

N

�� � ���� d�l��

N

���� �

N

�� � ���� d��

N

�� ��

d�k � ��

N�d�d � ��k�k � ��

�N��O

��dkN

����

��

When d � �N and k �� N �

R�d� � ��d�k � ��

N�d�d � ��

N�

kXl��

�k � �� l�

��� d

N

���� d��

N

�� � ���� d�l��

N

���� �

N

�� � ���� l

N

�

� �� �k � ��� � ��kXl��

�k � �� l��� � ��l�� �O�k��N�

� ��� ��k�� �O�k��N��

�