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population genetics is concerned with the investigation of the genetic structure of populations , which is influenced by evolutionary factors such as mutation , selection , recombination , migration and genetic drift . for excellent reviews of the theoretical aspects of this field , see @xcite .    in this paper , the antagonistic interplay of mutation and selection shall be investigated , with mutation generating the genetic variation upon which selection can act . pure mutation selection models exclude genetic drift and are therefore deterministic models , and accurate only in the limit of an infinite population size ( for a review , see * ? ? ? a further simplification taken here is to consider only _ haploid _ populations , where the genetic material exists in one copy only in each cell . however , the equations used here to describe evolution apply as well to diploid populations without dominance .    for the modelling of the types considered , the _ sequence space approach _ is used , which has first been used by @xcite to model the structure of the dna , where individuals are taken to be sequences . here , the sequences shall be written in a two - letter alphabet , thus simplifying the full four - letter structure of dna sequences . in this approach , the modelling is based on the microscopic level , at which the mutations occur , hence the mutation process is fairly straightforward to model . however , the modelling of selection is a more challenging task , as selection acts on the phenotype , and the mapping from genotype to phenotype is by no means simple . to this end , the concept of the _ fitness landscape _ @xcite is introduced as a function on the sequence space , assigning to each possible genotype a fitness value which determines the reproduction rate . apart from the problem that a realistic fitness landscape would have to be highly complex ( too complex for a mathematical treatment ) , there is also very limited information available concerning the nature of realistic fitness functions . therefore , the modelling of fitness is bound by feasibility , trying to mimic general features that are thought to be essential for realistic fitness landscapes such as the degree of ruggedness . a very common type of fitness functions is the class of permutation - invariant fitness functions , where the fitness of a sequence is determined by the number of mutations it carries compared to the wild - type , but not on the locations of the mutations within the sequence . although this model describes the accumulation of small mutational effects surprisingly well , it is a simplistic model that lacks a certain degree of ruggedness that is thought to be an important feature of realistic fitness landscapes @xcite .    in this paper , hopfield - type fitness functions @xcite are treated as a more complex model . here , the fitness of a sequence is not only determined by the number of mutations compared to one reference sequence , but to a number of predefined sequences , the _ patterns_. this yields a class of fitness landscapes that contain a higher degree of ruggedness , which can be tuned by the number of patterns chosen . while this can still be treated with the methods used here , it is a definite improvement on the restriction of permutation - invariant fitness functions . particular interest is taken in the phenomenon of mutation driven error thresholds , where the population in equilibrium changes from viable to non - viable within a narrow regime of mutation rates . in this paper , a few examples of hopfield - type fitness functions are investigated with respect to the error threshold phenomenon . section [ the mutation  selection model in sequence space ] introduces the basic mutation  selection model with its main observables . in section [ sequences as types ] , the model is applied to the sequence space approach , formulating the mutation and fitness models explicitly . sections [ lumping for the hopfield - type fitness ] and [ the maximum principle ] present the method , which relies on a lumping of the large number of sequences into classes on which a coarser mutation  selection process is formulated . this lumping is necessary to formulate a simple maximum principle to determine the population mean fitness in equilibrium . in section [ error thresholds ] , this maximum principle is used to investigate some examples of hopfield - type fitness functions with respect to the error threshold phenomenon . the model used here ( as detailed below ) is a pure mutation  selection model in a time - continuous formulation as used by @xcite and @xcite , for instance . [ [ population . ] ] population . + + + + + + + + + + +    the evolution of a population where the only evolutionary forces are mutation and selection is considered , thus excluding other factors such as drift or recombination for instance . individuals in the population shall be assigned a type @xmath0 from the finite _ type space _ @xmath1 . the population at any time @xmath2 is described by the _ population distribution _ @xmath3 , a vector of dimension @xmath4 , the cardinality of the type space . an entry @xmath5 gives the fraction of individuals in the population that are of type @xmath0 . thus the population is normalised such that @xmath6 . [ [ evolutionary - processes . ] ] evolutionary processes . + + + + + + + + + + + + + + + + + + + + + + +    the evolutionary processes that occur are birth , death and mutation events . birth and death events occur with rates @xmath7 and @xmath8 that depend on the type @xmath0 of the individual in question , and taken together , they give the effective reproductive rate , or _ fitness _ as @xmath9 . mutation from type @xmath0 to type @xmath10 depends on both initial and final type and happens with rate @xmath11 . these rates are conveniently collected in square matrices @xmath12 and @xmath13 of dimension @xmath4 , where the reproduction or fitness matrix @xmath12 with entries @xmath14 is diagonal . the off - diagonal entries of the mutation matrix @xmath13 are given by the mutation rates @xmath11 , and as mutation does not change the number of individuals , the diagonal entries of @xmath13 are chosen such that @xmath15 , which makes @xmath13 a markov generator . the time evolution operator @xmath16 is given by the sum of reproduction and mutation matrix , @xmath17 . [ [ deterministic - evolution - equation . ] ] deterministic evolution equation . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +    in the deterministic limit of an infinite population size , the evolution of the population is governed by the evolution equation @xmath18{\boldsymbol{p}}(t ) \;,\ ] ] where @xmath19 is the population mean fitness . the term with @xmath20 is needed to preserve the normalisation of the population . note that this term makes the evolution equation ( [ evolution equation ] ) nonlinear . [ [ equilibrium . ] ] equilibrium . + + + + + + + + + + + +    the main interest focuses on the equilibrium , i.e. , the behaviour if @xmath21 , which is attained for @xmath22 . all equilibrium quantities shall be denoted by omitting the argument @xmath2 , for instance the equilibrium population distribution is @xmath23 . in equilibrium , the evolution equation ( [ evolution equation ] ) becomes an eigenvalue equation for @xmath16 , with leading eigenvalue @xmath20 and corresponding eigenvector @xmath23 . if @xmath13 is irreducible , as shall be assumed throughout , perron - frobenius theory ( see , for instance , * ? ? * appendix ) applies , which guarantees that the leading eigenvalue @xmath20 of @xmath16 is non - degenerate and the corresponding right eigenvector @xmath23 is strictly positive , which implies that it can be normalised as a probability distribution .    [ [ ancestral - distribution . ] ] ancestral distribution . + + + + + + + + + + + + + + + + + + + + + + +    similarly to the population distribution @xmath23 , there is also another important distribution in this model , namely the ancestral distribution @xmath24 . consider the population at time @xmath25 , but count each individual not as its current type , but as the type its ancestor had at time @xmath2 . thus an entry of the ancestral distribution @xmath26 determines the fraction of the population at @xmath25 whose ancestor at time @xmath2 was of type @xmath0 . in the limit @xmath27 , this also approaches an equilibrium distribution @xmath28 . as shown by @xcite , the equilibrium ancestral distribution can be obtained as a product of the left and right pf ( perron - frobenius ) eigenvectors @xmath29 and @xmath23 of the time - evolution operator @xmath16 as @xmath30 , where @xmath29 is normalised such that @xmath31 .    [ [ population - and - ancestral - means . ] ] population and ancestral means . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +    any function on the type space given , say , by @xmath32 can be averaged with respect to the population or the ancestral distribution distribution . the population mean of @xmath33 is given by @xmath34 whereas the ancestral mean is @xmath35 note that the time - dependence of the means only comes from the distribution , whereas the function @xmath33 is considered constant in time . in equilibrium , time dependence is again omitted such that the equilibrium population and ancestral means are denoted @xmath36 and @xmath37 , respectively . an important example of the population mean is the population mean fitness @xmath38 from equation ( [ evolution equation ] ) . in the previous section , the types are a rather abstract concept . in order to formulate the particular mutation and fitness models , they shall now be specified as sequences , mimicking the structure of the dna ( cf . * ) . for simplicity , only two - state sequences are considered , i.e. , sequences that have at each site one out of two possible entries . however , the method used here can immediately be generalised to a more realistic four - state model ( see * ? ? ? * ) . the types therefore are associated with sequences @xmath39 of fixed length @xmath40 , written in the _ alphabet _ @xmath41 , thus @xmath42 for @xmath43 . this means that there are @xmath44 different sequences , and thus the type space ( or _ sequence space _ ) @xmath1 has cardinality @xmath45 . a simple mutation model that neglects any processes changing the length of the sequence , such as deletions or insertions , is used . mutations are modelled as point processes , where an arbitrary site @xmath46 is switched with rate @xmath47 , such that the mutation rate between sequences that differ only in one particular site is given by @xmath48 . sequences that differ in more than one site can not mutate into one another within a single mutational step . this is known as the _ single step mutation model _ , introduced by @xcite . the mutation model defines a neighbourhood in the sequence space @xcite . a convenient measure for the distance between sequences is the hamming distance @xmath49 , which counts the number of sites at which the sequences @xmath50 and @xmath51 differ @xcite . with this , the mutation matrix is explicitly given as @xmath52 the diagonal entry is chosen such that @xmath13 fulfils the markov condition @xmath53 . a rather simple , though commonly used type of fitness function is the _ permutation - invariant fitness_. there , the fitness of a sequence depends only on the number of mutations it has compared to a reference type , not on their position along the sequence . thus fitness is a function of the hamming distance to the reference sequence , which is usually chosen as the wild - type . the hamming distance to the wild - type is also called the _ mutational distance _ @xmath54 .    a non - permutation - invariant fitness that contains some ruggedness , but is simple enough to be dealt with in this framework , is the _ hopfield - type fitness _ , a special type of spin - glass model , which has been introduced by @xcite as a model for neural networks . instead of comparing a sequence only to the wild - type , as it is done for the permutation - invariant fitness , the hopfield - type fitness of a sequence is determined by its hamming distances to @xmath55 reference sequences , the _ patterns _ @xmath56 , @xmath57 . the hopfield - type fitness shall be defined in terms of the _ specific distances _ @xmath58 , which are the hamming distances with respect to the patterns , @xmath59 and thus the fitness is given as @xmath60 note that in the case of a single pattern ( @xmath61 ) , this yields again a permutation - invariant fitness . one problem of the sequence space approach is the large number of types , which grows exponentially with the sequence length @xmath40 , @xmath45 . the time - evolution operator @xmath16 is a matrix of size @xmath62 , and in this set - up one is interested in its leading eigenvalue @xmath20 and the corresponding right and left eigenvectors @xmath23 and @xmath29 . the relevant sequence length depends on the particular application one has in mind , but it is typically rather long . if one aims to model the whole genome of a virus or a bacterium , @xmath40 has to be in the region of @xmath63 , but even a single gene has of the order of @xmath64 base pairs . these values lead to matrices of a size that makes the eigenvalues and eigenvectors inaccessible .    for some types of fitness functions , this problem can be reduced by _ lumping _ together types into _ classes _ of types , and considering the new process on a reduced sequence space , which contains the classes rather than the individual types . under certain circumstances , mutation is described as a markov process in the emerging lumped process as well , such that this process is accessible to markov process methods , and the framework developed in section [ the mutation  selection model in sequence space ] can directly be applied to the lumped system . the lumping of the mutation process is a standard procedure in the theory of markov chains @xcite , see also @xcite for an application to mutation  selection models . this lumping leads to a meaningful mutation  selection model on the reduced type space , if all sequences lumped together into one class have the same fitness . it is possible to lump the markov chain given by the mutation matrix @xmath13 with state space @xmath1 with respect to a particular partition @xmath65 , if and only if for each pair @xmath66 the cumulative mutation rates @xmath67 from type @xmath68 into @xmath69 , are identical for all @xmath68 , cf .  the example shown in figure [ fig lumping ] . ( 258,171 ) ( 15 , 5)(0,0)@xmath70 ( 200 , 30)(0,0)@xmath69 ( 23 , 90)(0,0)@xmath71 ( 25 , 40)(0,0)@xmath72 visualisation of the compatibility with lumping : consider two classes @xmath70 and @xmath69 . the mutation rates from the types in @xmath70 to the types in @xmath69 ( given next to the arrows ) are compatible with a lumping with respect to @xmath70 and @xmath69 , because the sum of the mutation rates from type @xmath71 to all types in @xmath69 is given by @xmath73 , which is identical with those from @xmath72 , @xmath74 . , title="fig:",scaledwidth=60.0% ]    in this case , the lumped process , with @xmath75 states @xmath76 and mutation rates @xmath77 for any @xmath68 , is again a markov chain ( * ? ? ? * theorem 6.3.2 ) .      whereas in the case of a permutation - invariant fitness function , the lumping procedure is fairly simple , collecting all sequences with the same hamming distance to the wild - type into classes , and considering cumulative mutation rates between these classes , the lumping for the hopfield - type fitness is somewhat more complex . for a two - state model with hopfield fitness , it has been performed for instance by @xcite , and this shall be recollected in the remainder of this section . first , the quantities with respect to which the lumping shall be performed must be defined . to this end , consider as an example the case of sequence length @xmath78 with three patterns ( @xmath79 ) , and let the patterns @xmath56 be collected in a @xmath80 matrix @xmath81 , such that the @xmath82th row of @xmath81 is pattern @xmath56 . without loss of generality , the pattern @xmath83 can always be chosen as @xmath84 . let the patterns in this example be given as @xmath85 note that there are only @xmath86 different types of sites ( corresponding to the columns of @xmath81 ) . these are collected in a @xmath87 matrix @xmath88 , the columns of which correspond to the possible types of sites in the matrix of patterns @xmath81 . for the case @xmath79 , this @xmath89 matrix is given as @xmath90 using the column vectors @xmath91 of the matrix @xmath88 , the patterns given in equation ( [ example pattern ] ) can alternatively be expressed as @xmath92 classifying the sites into @xmath86 classes according to which of the column vectors @xmath91 of @xmath88 coincides with the column vector @xmath93 of the patterns @xmath81 at site @xmath94 . let @xmath95 be the index set of sites , with a partition into @xmath86 subsets @xmath96 induced by the patterns @xmath81 such that @xmath97 with @xmath98 patterns can be characterised by the number of sites @xmath99 in each subset . the example patterns ( [ example pattern ] ) can therefore be described by @xmath100    considering only subsequences @xmath101 , the _ partial distances _ @xmath102 of a sequence @xmath50 with respect to the pattern @xmath83 are defined as the hamming distance between @xmath50 and @xmath83 , _ restricted to the subsequence @xmath103_. therefore , they can be written as @xmath104 such that the specific distance with respect to pattern @xmath83 is given by @xmath105 .    because the differences between each of the patterns @xmath56 within the subsets @xmath96 of @xmath106 are known ( and recorded in the matrix @xmath88 ) , it is sufficient to consider only the partial distances @xmath102 with respect to one pattern , here @xmath83 ; the partial distances @xmath107 with respect to any other pattern can be expressed in terms of the @xmath102 as @xmath108 using the matrix elements @xmath109 of @xmath88 . the specific distance @xmath58 to any pattern @xmath56 can be expressed as @xmath110 where the index set @xmath111 of classes is partitioned into two subsets , @xmath112 with @xmath113 and @xmath114 . hence , by specifying the @xmath86 partial distances @xmath102 with respect to pattern @xmath83 , the specific distances @xmath58 with respect to any pattern @xmath56 are determined , which in turn determine the fitness . this implies that all sequences with the same partial distances @xmath102 have the same fitness . thus the partial distances @xmath102 to pattern @xmath83 , collected in a _ mutational distance _ vector @xmath115 , shall be the quantities that label the classes in the lumped system .      the relevant partition of the sequence space is given by @xmath116 with @xmath117 , and the reduced sequence space , or _ mutational distance space _ @xmath118 , contains the classes @xmath119 . considering again the subsequences @xmath120 , there are @xmath121 possible different @xmath102 ( as @xmath102 takes values from @xmath122 to @xmath123 ) , and there are @xmath124 different subsequences @xmath103 for each @xmath102 . hence , considering all sites , there are @xmath125 , and @xmath126 sequences @xmath50 that are mapped onto each @xmath127 . for the patterns @xmath81 chosen as example ( [ example pattern ] ) , we have @xmath128 , while the full sequence space has dimension @xmath129 .    in the single step mutation model , the only neighbours of a sequence @xmath50 with distance vector @xmath127 lie in the classes @xmath130 , where the @xmath131 are the unit vectors of mutation . thus the only non - zero cumulative mutation rates are @xmath132 as a sequence with @xmath127 has @xmath133 @xmath122-sites and @xmath102 @xmath134-sites in @xmath96 , and the single - site mutation rates are @xmath48 for all sites , the cumulative mutation rates are given by @xmath135 irrespective of the particular order within the subsequences @xmath103 . therefore the cumulative mutation rates @xmath136 are the same for all sequences @xmath50 with the same @xmath127 , which is the condition for lumping . the mutation  selection model with hopfield - type fitness is indeed `` lumpable '' with respect to the partition induced by the distance vectors @xmath127 .    the mutation selection process on the mutational distance space @xmath118 is described by the lumped time - evolution operator @xmath137 with lumped reproduction and mutation matrices @xmath138 and @xmath139 of dimension @xmath140 . whereas the lumped reproduction matrix @xmath138 is still diagonal and contains the same entries as @xmath12 , i.e.  @xmath141 , the off - diagonal entries of the lumped mutation matrix @xmath139 are given by the cumulative mutation rates with unchanged diagonal entries compared to @xmath13 , which still fulfil the markov property @xmath142 , and thus @xmath143 with the cumulative mutation rates from equation ( [ cumulative mutation rates ] ) . for a more general derivation of the lumped reproduction and mutation matrices , see @xcite . note that the time - evolution operator @xmath144 acting on @xmath118 describes the evolution of a population under mutation and selection determined by the evolution equation ( [ evolution equation ] ) , and thus the theory developed in section [ the mutation  selection model in sequence space ] applies . although the lumping procedure reduces the number of types very efficiently , the evaluation of the eigenvalues and eigenvectors of the time - evolution operator @xmath144 still remains a difficult problem for many applications , due to the size of the eigenvalue problem . if one is interested solely in the equilibrium behaviour of the system , however , it is possible to determine the population mean fitness ( at least asymptotically for large sequence length @xmath40 ) , given by the leading eigenvalue of @xmath144 . this can be done by a simple maximum principle that can be derived from rayleigh s general maximum principle , which specifies that the leading eigenvalue @xmath145 of an @xmath146 matrix @xmath144 can be obtained via a maximisation over @xmath147 , @xmath148 the vector @xmath149 for which the supremum is attained is the eigenvector corresponding to the eigenvalue @xmath145 . the simple maximum principle derived from this guarantees that the population mean fitness @xmath20 can be obtained by maximising a function on the mutational distance space @xmath118 . it can be shown that the maximiser itself is the ancestral mean mutational distance . such a maximum principle has first been derived by @xcite for two - state sequences with permutation - invariant fitness . this has been generalised to apply for four - state sequences with permutation - invariant fitness by @xcite , and subsequently the restriction to permutation - invariant fitness function has been relaxed by @xcite . the results from @xcite apply directly to the hopfield - type fitness treated here .      whereas the original mutation matrix @xmath13 is symmetric , the lumped mutation matrix @xmath139 is no longer symmetric , as different numbers of sequences are lumped into the different classes , therefore giving rise to unequal cumulative forward and backward mutation rates . to derive the maximum principle , it is necessary to symmetrise the mutation matrix @xmath139 . @xmath139 is reversible , i.e. , @xmath150 where @xmath151 is the stationary distribution of the pure mutation process , which is given by the equidistribution of types on @xmath1 , and thus given by the number of sequences @xmath152 that are lumped onto the same mutational distance vector @xmath127 . the reversibility of @xmath139 implies that it can be symmetrised by the means of a diagonal transformation @xmath153 , which yields the symmetrised mutation matrix as @xmath154 with off - diagonal entries @xmath155 using the cumulative mutation rates @xmath156 , this reads @xmath157 as can be seen by using the explicit representation for the cumulative mutation rates from equation ( [ cumulative mutation rates ] ) with the @xmath152 from equation ( [ nd ] ) . because @xmath158 is diagonal , the diagonal entries of the mutation matrix are unchanged , @xmath159    as @xmath138 is diagonal as well , it is not changed by the transformation @xmath160 , and thus this transformation also symmetrises the time - evolution operator such that @xmath161 is symmetric .    before symmetrisation , @xmath144 was expressed as the sum of a markov generator @xmath139 and a diagonal remainder @xmath138 . as the transformation @xmath158 does not preserve the markov property , this is not the case for the symmetrised time - evolution operator in ( [ symmetrised time - evolution operator ] ) . it is however useful to split it up this way . to this end , let @xmath162 where @xmath163 is a ( symmetric ) markov generator and @xmath164 is the ( diagonal ) remainder . the off - diagonal entries of @xmath163 are given by those of @xmath165 from equation ( [ symmetrised mutation rates ] ) , @xmath166 for @xmath167 , whereas the markov property requires as diagonal entries @xmath168 the remainder @xmath164 is given by @xmath169      to deal with the case of infinite sequence length , it will prove useful to use intensively scaled normalised versions of the extensively scaled variables like the mutational distances . the pattern in the hopfield model , previously characterised by the _ number _ of sites @xmath123 in each subset @xmath96 , will now be described by the _ fraction _ of sites in @xmath96 , given by @xmath170 . similarly , we use normalised partial distances @xmath171 where @xmath172 $ ] , with the normalised mutational distance vector @xmath173 . the permutation - invariant model is obtained in the case @xmath61 .    for finite @xmath40 , @xmath174 takes rational values in a normalised version of the mutational distance space @xmath175 , where @xmath176 is a compact domain in @xmath177 . for @xmath178 , the vectors @xmath174 become dense in @xmath176 . assume that the entries of @xmath179 can be approximated by functions @xmath180 and @xmath33 from @xmath181 , i.e. , twice continuously differentiable functions with bounded second derivatives that map @xmath176 onto @xmath182 such that @xmath183 where @xmath184 and the notation @xmath185 is used to emphasise that the normalised mutational distance @xmath174 corresponding to a particular @xmath127 is meant . in fact , assumption ( [ assumption f ] ) can readily be verified for the cumulative mutation rates from equation ( [ cumulative mutation rates ] ) :    let the functions @xmath186 be @xmath187 where the functions @xmath188 are given by @xmath189 with the cumulative mutation rates @xmath190 , which read explicitly @xmath191 using a taylor approximation , it can be shown that the differences between the exact entries of @xmath192 as given in equations ( [ symmetrised mutation rates ] ) and ( [ diagonal entries of f ] ) , and their approximations from equation ( [ function f ] ) , are indeed of @xmath193 . assuming that also the reproduction rates @xmath194 can be approximated by a @xmath195 function @xmath196 as @xmath197 then from equation ( [ remainder e ] ) , the matrix @xmath164 is approximated by @xmath198 fulfilling equation ( [ assumption e ] ) . with the definition of the _ mutational loss function _ @xmath199 as @xmath200 \,,\end{aligned}\ ] ] this reads explicitly @xmath201    we are now in a position to apply theorems 1 and 2 from @xcite , which read for the mutation  selection model with hopfield - type fitness considered here     +   _ ( i ) assume that for the lumped mutation  selection model as set up in section [ lumping for the hopfield - type fitness ] it is possible to approximate the reproduction rates @xmath194 by a @xmath195 function @xmath202 as specified in equation ( [ assumption r ] ) , and that the @xmath195 function @xmath180 assumes its global maximum in the interior _ @xmath203_. then the population mean fitness in equilibrium is given by @xmath204 + { \mathcal{o}\left(\frac{1}{n}\right)}\,.\ ] ] ( ii ) assume furthermore that @xmath180 assumes its maximum at a unique point @xmath205 in _ @xmath203 _ , and that the hessian of @xmath180 at @xmath205 is negative definite . then in the limit of @xmath178 , the maximiser @xmath205 is given by the mean ancestral mutational distance @xmath206 , and in particular @xmath207 _ for a proof the reader is referred to @xcite . the maximum principle ( [ maximum principle ] ) and ( [ ancestral maximum principle ] ) is a powerful tool to calculate the population mean fitness @xmath20 in equilibrium for arbitrary fitness functions of the permutation - invariant or hopfield type , for any range of mutation rates . also , the ancestral mean genotype @xmath206 is available . the general method to identify @xmath20 and @xmath206 is to consider the partial derivatives of @xmath208 with respect to the components @xmath209 of the mutational distance @xmath174 . a necessary condition for the function @xmath208 to have a maximum at a value @xmath205 is that its derivatives at this @xmath205 vanish , @xmath210_{{\boldsymbol{x}}={\boldsymbol{x}}^ * } = 0   \quad\forall\ , v , k \;.\ ] ] the global maximum of the function @xmath208 must lie on one of the points @xmath205 that fulfil equation ( [ error threshold general method ] ) or on the border of the mutational distance space . thus by comparing the values of @xmath208 on these possible points , the global maximum can be identified .    apart from the general possibility to investigate the dependence of the population mean fitness @xmath20 on the mutation rate @xmath47 , this yields the opportunity to investigate the phenomenon of the _ error threshold _ , which has interested scientists ever since it was first conceived by @xcite . the phenomenon of the error threshold can be described as the existence of a critical mutation rate , below which the equilibrium population is well localised in sequence space , whereas for mutation rates above the critical mutation rate , the equilibrium population is more delocalised , with a sharp transition between the two phases . one problem is that there is no generally accepted definition of an error threshold . the criterion used in the original quasispecies model @xcite is the disappearance of the wild - type from the population , which under the single peaked landscape used there goes in line with the complete delocalisation of the population in sequence space . however , these two effects do not necessarily coincide for other fitness landscapes . the definition of the error threshold that shall be used here is equivalent to the definition of a phase transition in physics , differentiating between first and second order transitions as follows :    * definition ( first and second order error threshold ) . * + ( i ) a first order error threshold exists at a critical mutation rate @xmath211 , if the ancestral mean mutational distance as a function of the mutation rate @xmath212 shows a discontinuity at this @xmath211 , which is also reflected by a kink in the population mean fitness @xmath213 . + ( ii ) a second order error threshold exists at a critical mutation rate @xmath211 , if the ancestral mean mutational distance is continuous , but its derivative with respect to the mutation rate @xmath214_{\mu\rightarrow\mu_c}$ ] is discontinuous at this mutation rate @xmath211 .    in the examples shown later in this thesis , the second order error threshold always show an infinite derivative at the critical mutation rates . note that , like phase transitions in physics , these definitions of the error thresholds apply in the strict sense only to a system with infinite sequence length ( @xmath178 ) , for finite sequence lengths , the thresholds are smoothed out due to the lack of non - analyticities . @xcite gave a finer classification of different error threshold phenomena . the first order error threshold they called `` fitness threshold '' . here , this term shall include also the second order error threshold , making all error thresholds as defined above fitness thresholds . furthermore , the concept of the `` degradation threshold '' was introduced :    * definition ( degradation threshold ) . * + a degradation threshold is an error threshold of first or second order , where the population distribution beyond the critical mutation rate @xmath211 is given by the equidistribution in sequence space @xmath1 . thus here the degradation threshold is a special case of a fitness threshold , going in line with the complete delocalisation of the population in sequence space . note that in the limit of infinite sequence length ( @xmath178 ) , for which the error threshold definitions apply exactly , this equidistribution is reached immediately above @xmath211 , and beyond the threshold the population is insensitive to any further increase in mutation rates . in the case of finite sequence lengths , where the thresholds are smoothed out , the equidistribution is of course only reached asymptotically . the original error threshold was observed for the single peaked fitness landscape , where a single sequence is attributed a high fitness value , all other sequences are equally disadvantageous ( for a review , see * ? ? ? this is clearly an oversimplification and should not be regarded as anything but a toy model . other fitness landscapes that have been investigated comprise , in the permutation - invariant case , linear and quadratic fitness functions , general functions showing epistasis , and as examples lacking permutation - invariance the onsager landscape @xcite , which has nearest neighbour interactions within the sequence , as well as various spin glass landscapes like the hopfield landscape @xcite , the sherrington - kirkpatrick spin glass @xcite , the nk spin glass @xcite , and the random energy model @xcite , assigning random fitness values to each sequence . one fitness landscape where an analytical solution can be obtained is the linear fitness ( cf . * ? ? ? * ; * ? ? ? * ; * ? ? ? note that this corresponds to a multiplicative landscape in a set - up using discrete time . for a linear fitness function , there is no error threshold , but the population changes smoothly from localised to delocalised with an increasing mutation rate .    for quadratic fitness functions , error thresholds only exist for antagonistic epistasis ; they are absent for quadratic fitness functions with synergistic epistasis @xcite . these results go in line with those for general epistatic fitness functions @xcite . studies using non - permutation - invariant fitness functions generally report the presence of error thresholds . of course the discussion of the error threshold phenomenon is academic if the threshold is an artifact of the model rather than a real biological phenomenon . this issue has been subject to numerous debates , especially because it has first been predicted by a model using the over - simplistic single peaked landscape . however , over the years biologists have accumulated evidence that particularly rna viruses naturally thrive at very high mutation rates @xcite , of the order of @xmath215 to @xmath216 per base per replication @xcite , corresponding to a genomic mutation rate of about 0.1 to 10 mutations per replication @xcite , and a number of studies have reported that populations of rna viruses only survive a moderate increase of their mutation rate , whereas if the mutation rate is increased further , the populations become extinct @xcite , for reviews see @xcite . this corresponds to the population being pushed beyond the error threshold . it has been suggested to use the error threshold for anti - viral therapies @xcite , and in fact , recent experimental results indicate that this is the mechanism via which the broad - spectrum anti - viral drug ribavirin works @xcite . this clearly warrants some further investigation of the error threshold phenomenon , which shall be done in the remainder of this section . the original hopfield fitness as introduced by @xcite is a quadratic function of the specific distances and reads @xmath217 using normalised specific distances @xmath218 , similarly to the normalised mutational distances @xmath174 . the statistical properties of this landscape have been studied in detail @xcite : in the thermodynamic limit @xmath219 , there are @xmath220 global maxima that are associated with the patterns @xmath221 and their complements @xmath222 . in addition to that , the number of local maxima and saddle points grows exponentially with the number patterns @xmath55 , hence the ruggedness of the fitness landscape can be tuned by the number of patterns . most works that have studied a hopfield - type fitness used the original hopfield model , a generalisation was however treated by @xcite , using a hopfield - type truncation selection with two patterns . thus it might be interesting and instructive to investigate the threshold behaviour of different kinds of hopfield - type fitness functions . applying criteria for the existence of error thresholds that have been obtained by @xcite for permutation - invariant fitness functions to the case of a hopfield - type fitness , it can be shown that for linear hopfield - type fitness functions there are no error thresholds , which is a new result , considering that for all previously investigated hopfield - type fitness functions , the existence of error thresholds was reported . the next step towards more complex fitness functions is to consider quadratic fitness functions , generalising the original hopfield - fitness , which is a particular example for a quadratic function . here , the analysis shall be restricted to a symmetry with respect to the normalised specific distances @xmath223 to the patterns @xmath56 , such that @xmath224 the parameter @xmath225 tunes the linear in relation to the quadratic term , and the sign of the quadratic term determines the _ epistasis _ , a measure for the strength of interaction between sites . for a positive quadratic term epistasis is said to be negative or antagonistic , whereas for a negative quadratic term one speaks of positive or synergistic epistasis . the case @xmath226 combined with a positive quadratic term ( i.e. , negative epistasis ) yields the original hopfield fitness .      in the case of two patterns , @xmath227 , the first pattern can be chosen without loss of generality as @xmath228 , such that there is only one pattern to be chosen , usually randomly . the matrix @xmath88 containing the possible types of sites is given by @xmath229 and thus the index set of sites is partitioned into two subsets , @xmath230 , where @xmath231 contains all sites at which both patterns have entry @xmath122 , whereas @xmath232 corresponds to the sites where the two patterns have entries @xmath122 and @xmath134 , respectively . the only quantities characterising the patterns are now the fractions of sites in each partition , @xmath233 and @xmath234 . thus the pattern can be characterised by a single parameter , @xmath235 . each sequence is characterised with respect to the pattern by the partial hamming distances to pattern @xmath83 ( in normalised form ) , @xmath236 and @xmath237 . these vary from @xmath122 ( all entries @xmath122 in @xmath96 ) to @xmath134 ( all entries @xmath134 in @xmath96 ) , completely independently from each other . the specific distances with respect to the patterns are linear combinations of the @xmath209 and given in normalised form by @xmath238 the hopfield - type fitness is defined as an arbitrary function of these patterns , @xmath239 . due to the small number of variables , for the case of two patterns , a lot can be done by analytical treatment .    for the quadratic symmetric hopfield - type fitness ( [ quadratic symmetric hopfield - type fitness ] ) with positive epistasis ( i.e. , negative sign of the quadratic term ) and @xmath240 , there are no phase transitions , going in line with the results for permutation - invariant fitness functions , but different from other results for hopfield - type fitness functions . as an example for negative epistasis , consider first the original hopfield fitness ( [ original hopfield fitness ] ) . since for two patterns there are only two variables , it is possible to visualise the fitness landscape in this case . figure [ fig 6.14 ] shows the original hopfield fitness ( [ original hopfield fitness ] ) for the cases @xmath241 and @xmath242 .    original hopfield fitness in the case of two patterns with different @xmath243.,title="fig:",scaledwidth=45.0% ] original hopfield fitness in the case of two patterns with different @xmath243.,title="fig:",scaledwidth=45.0% ]    in the corners of the mutational distance space @xmath118 , one can see the four degenerate maxima . the ancestral mean partial distances @xmath244 , at which the maxima of @xmath208 are positioned , are obtained by considering the derivatives of @xmath208 . they are given by @xmath245 for the case of @xmath241 , which corresponds to two completely uncorrelated patterns , the ancestral mean partial distances @xmath244 are shown in figure [ fig 6.15 ] on the left , alongside the ancestral mean specific distances @xmath246 .    ancestral mean partial distances @xmath244 ( top ) and specific distances @xmath247 ( bottom ) depending on mutation rates . the original hopfield fitness ( [ original hopfield fitness ] ) for two patterns has been used . results correspond to uncorrelated patterns ( @xmath241 , left ) and correlated patterns with with @xmath242 ( right ) . ]    for low mutation rates , there are two possible solutions for each of the ancestral mean partial distances @xmath244 , and as the maxima are degenerate , in equilibrium , the population will be centred equally around all of them . however , in the approach to equilibrium , the population might well be predominantly concentrated around one of them , depending on initial conditions . the specific distances @xmath223 that are shown correspond to the combination of @xmath248 and @xmath249 , where both are given by the lower branch . other combinations yield similar results . for high mutation rates , the population is in the mutation equilibrium with @xmath250 , forming a disordered phase . in the limit of low mutation rates , @xmath251 , the population is always in the vicinity of one of the patterns ( or its complement ) , such that one of the @xmath252 , which is completely random with respect to the other pattern , and thus the other @xmath253 . this is the ordered phase . at the critical mutation rate @xmath254 , there is a second order phase transition between these two phases , which is a fitness as well as a degradation threshold , corresponding to the infinite derivative of both @xmath244 at this mutation rate . as the specific distances @xmath223 are simply superpositions of the partial distances @xmath209 , the phase transitions are also visible in the @xmath223 .    in the correlated case @xmath255 ( figure [ fig 6.15 ] , right ) , two second order transitions can be identified . at @xmath256 , @xmath248 has a phase transition , whereas at @xmath257 , @xmath249 has a phase transition . the threshold occurring at the lower mutation rate is only a fitness threshold , whereas the one happening at the higher mutation rate is both a fitness and degradation threshold , leading to a totally random population . for @xmath258 , the population is in an ordered phase , for @xmath259 , it is in a partially ordered phase , which is ordered with respect to one of the variables , but random with respect to the other . finally , for @xmath260 , the population is the equidistribution in sequence space . here again , for low mutation rates the population is close to one of the patterns , but due to the correlation in the chosen patterns , this leads to a non - random overlap with the other pattern . in the uncorrelated case with @xmath241 , the two error thresholds coincide , and the partially ordered phase vanishes . now turn to the question how these phase transitions depend on the particular degeneracy of the fitness functions and consider the quadratic fitness function ( [ quadratic symmetric hopfield - type fitness ] ) with negative epistasis ( i.e. , positive quadratic term ) for values of @xmath261 . figure [ fig 6.16 ] shows the fitness landscapes for values of @xmath262 and @xmath263 for uncorrelated patterns @xmath241 and correlated patterns with @xmath242 . quadratic hopfield - type fitness functions ( [ quadratic symmetric hopfield - type fitness ] ) with negative epistasis and @xmath262 ( top ) and @xmath263 ( bottom ) for an uncorrelated pattern @xmath241 ( left ) and a correlated pattern with @xmath242 ( right).,title="fig:",scaledwidth=47.0% ] quadratic hopfield - type fitness functions ( [ quadratic symmetric hopfield - type fitness ] ) with negative epistasis and @xmath262 ( top ) and @xmath263 ( bottom ) for an uncorrelated pattern @xmath241 ( left ) and a correlated pattern with @xmath242 ( right).,title="fig:",scaledwidth=43.0% ]    quadratic hopfield - type fitness functions ( [ quadratic symmetric hopfield - type fitness ] ) with negative epistasis and @xmath262 ( top ) and @xmath263 ( bottom ) for an uncorrelated pattern @xmath241 ( left ) and a correlated pattern with @xmath242 ( right).,title="fig:",scaledwidth=47.0% ] quadratic hopfield - type fitness functions ( [ quadratic symmetric hopfield - type fitness ] ) with negative epistasis and @xmath262 ( top ) and @xmath263 ( bottom ) for an uncorrelated pattern @xmath241 ( left ) and a correlated pattern with @xmath242 ( right).,title="fig:",scaledwidth=43.0% ]    as the pictures indicate , the fitness functions ( and thus the behaviour of the system ) with the same @xmath264 are related by symmetry under @xmath265 ( apart from a constant term , which does not influence the dynamics ) . note that in @xmath237-direction , the fitness function is independent of @xmath225 . this is because in the sum of the specific distances , the term with @xmath237 cancels out , which happens generally in the case of an even number of patterns ( i.e. , odd @xmath266 ) for @xmath267 different variables .    because in @xmath237-direction the fitness is independent of @xmath225 , the solution for the ancestral mean mutational distance @xmath249 is identical with the solution for the original hopfield fitness as given in equation ( [ two - state two - pattern analytical solution ] ) . so for all values of @xmath225 , the phase transition with respect to @xmath237 happens at @xmath257 . for @xmath236 , the solution becomes more complicated , but the inverse function is simpler . it is given by @xmath268\sqrt{\hat{x}_1(1-\hat{x}_1 ) } } { 2\hat{x}_1 - 1 } \,.\ ] ] the dependence of @xmath248 on the mutation rate is shown in figure [ fig 6.17 ] ( top ) . ancestral mean partial distance @xmath248 ( top ) and specific distances @xmath269 ( thick lines , bottom ) and @xmath270 ( thin lines , bottom ) depending on mutation rates . the quadratic hopfield - type fitness ( [ quadratic symmetric hopfield - type fitness ] ) with negative epistasis for two patterns and different values of @xmath225 has been used . results correspond to uncorrelated patterns @xmath241 ( left ) and correlated patterns with @xmath242 ( right ) . data are shown for parameter values of @xmath271 ( top to bottom ) . for clarity , only specific distances @xmath247 corresponding to @xmath272 are shown . ]    for @xmath261 , the second order phase transition is smoothed out and thus vanishes . note that the ambiguity in the solutions that exists in the case @xmath226 ( cf . figure [ fig 6.15 ] ) , does not exist here , due to the lacking degeneracy of the maxima of the fitness function at @xmath273 and @xmath274 ( cf . figure [ fig 6.16 ] ) . at the bottom , figure [ fig 6.17 ] shows the specific distances @xmath247 , using the lower branch of the solution for @xmath249 ( as shown in figure [ fig 6.15 ] ) , which show the second order transition in @xmath249 , a fitness threshold . with this combination of solutions , for low mutation rates the population is centred around the sequence complementary to pattern @xmath275 . the general picture for the uncorrelated ( @xmath276 ) and correlated ( @xmath277 ) choice of patterns is very similar , apart from issues like the exact location of the thresholds . the behaviour of the system with quadratic symmetric hopfield - type fitness has been investigated for three , four and five patterns . however , due to the complexity of the analysis for a higher number of patterns , the focus is on the case of three patterns .    for three patterns , the matrix @xmath88 reads @xmath278 thus there are four @xmath279 describing the patterns @xmath81 , fulfilling @xmath280 , and four variables @xmath172 $ ] , describing each sequence . the specific distances with respect to pattern @xmath56 are given by @xmath281    similarly to the case of two patterns , the original hopfield fitness ( [ original hopfield fitness ] ) shall be considered first , and then variations ( [ quadratic symmetric hopfield - type fitness ] ) with negative epistasis , but @xmath261 . the investigation is done by means of numerical calculations .      for the hopfield - type fitness , the number of classes in the lumped system is given by @xmath282 . now consider the case where the patterns are chosen randomly with equal probability for each letter at each site . this results in a multinomial probability distribution for the number of sites @xmath123 in each subset @xcite , @xmath283 where @xmath284 . the means are given by @xmath285 and the variance @xmath286 such that @xmath287 . if the patterns @xmath56 for @xmath288 are chosen randomly ( remember @xmath228 ) , the @xmath279 are thus given by @xmath289 . so to mimic the case of an infinite sequence length , in which the maximum principle is exact , one has to assume uncorrelated patterns with @xmath290 for all @xmath291 . however , the maximum principle can also be used to investigate the case of finite sequence length by simulating the finite sequence length through choosing pattern distributions that do not follow exactly the infinite distribution @xmath290 , but vary around this mean value with a variance according to the sequence length to be simulated . practically , patterns corresponding to finite sequence length @xmath40 have been obtained by choosing for the @xmath292 sites entries @xmath122 or @xmath134 with probability 1/2 at each site , and counting the number of sites @xmath123 in each class @xmath96 , similarly to the example pattern given in equation ( [ example pattern ] ) . thus although the patterns were chosen randomly , they are correlated due to the finite sequence length .    as shall be seen in the following , these correlations do account for some interesting additional features . however , focus first on the case of a genuinely infinite sequence length with @xmath290 for all @xmath293 . [ [ original - hopfield - fitness - c-1 . ] ] original hopfield fitness ( @xmath226 ) . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +    the results for the case of three , four and five patterns with original hopfield fitness ( [ original hopfield fitness ] ) and uncorrelated patterns ( @xmath290 for all @xmath293 , corresponding to random patterns for infinite sequence length ) look exactly like those for two patterns shown in figure [ fig 6.15 ] ( left ) .    the solutions for the different @xmath209 all coincide . for small mutation rates @xmath294 , there are again two degenerate solutions for each @xmath209 , which can be combined in multiple ways for the different @xmath209 , yielding any of the patterns or their complementary sequences . at @xmath295 , there is a second order phase transition , which is a fitness and degradation threshold . for small mutation rates , the population is centred around one of the patterns , say @xmath296 , and therefore @xmath297 , whereas it is completely random with respect to the other patterns , yielding @xmath298 for @xmath299 . [ [ deviations - from - the - original - hopfield - fitness - cneq-1 . ] ] deviations from the original hopfield fitness ( @xmath261 ) . + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +    ancestral mean partial distances @xmath244 ( top ) and specific distances @xmath247 ( bottom ) depending on mutation rates . the quadratic hopfield - type fitness ( [ quadratic symmetric hopfield - type fitness ] ) with negative epistasis for three patterns with @xmath262 ( left ) and @xmath263 ( right ) has been used . results correspond to uncorrelated patterns for infinite sequence length ( @xmath300 ) . ] figure [ fig 6.19 ] shows the ancestral mean partial distances @xmath244 and specific distances @xmath247 for the quadratic hopfield - type fitness ( [ quadratic symmetric hopfield - type fitness ] ) with negative epistasis for different values of @xmath225 in the case of three patterns . data for four patterns look very similar ( not shown ) . as @xmath225 deviates from @xmath301 , the solutions for the @xmath209 do not coincide , and the phase transition becomes a first order fitness threshold , at which all four partial distances @xmath244 jump , but it is no more a degradation threshold . so contrary to the case of two patterns , where @xmath249 is independent of @xmath225 and the error threshold in @xmath248 is smoothed out by @xmath225 deviating from @xmath301 , here the threshold concerns all four partial distances @xmath244 and is sharpened to first order by @xmath261 .    for @xmath261 , the degeneracy between the patterns and their complements is lifted , and thus for mutation rates below the threshold , there are only @xmath55 different solutions , correlated with the patterns for @xmath302 , and with their complements for @xmath272 . for clarity , only one of the solutions is shown in figure [ fig 6.19 ] .    furthermore , the critical mutation rate decreases with increasing @xmath264 . the dependence of the critical mutation rate @xmath211 on the value of @xmath225 for @xmath303 is shown in figure [ fig 6.20 ] . the critical mutation rate @xmath211 depending on the value of @xmath225 . the quadratic hopfield - type fitness ( [ quadratic symmetric hopfield - type fitness ] ) with negative epistasis for three and four patterns has been used . results correspond to uncorrelated patterns simulating infinite sequence length.,scaledwidth=75.0% ]    at @xmath304 and @xmath305 , the critical mutation rate is @xmath306 , and for values of @xmath307 $ ] , there are no first order error thresholds for either @xmath79 or @xmath308 . this goes in line with a different sequence becoming optimal at @xmath309 for these values of @xmath225 . however , for @xmath308 , there is an additional line of second order error thresholds , that approaches @xmath310 as @xmath264 grows . preliminary results for @xmath311 indicate , that in that case the second order line does not occur . it might thus be conjectured that the existence of the second order error threshold line depends on the number of patterns being even or odd ( remember that for @xmath227 it does exist ) . this is an interesting result , as for all previously investigated hopfield - type fitness functions ( which are limited to the original hopfield fitness and a hopfield - type truncation selection as far as the author is aware ) , the existence of error thresholds has been reported . ancestral mean partial and specific distances , @xmath244 and @xmath247 , depending on mutation rates . the original hopfield fitness ( [ original hopfield fitness ] ) for three patterns has been used . results correspond to two typical examples of random , but correlated patterns chosen for sequences of lengths @xmath312 ( left ) , @xmath313 ( middle ) and @xmath314 ( right ) , specified at the top of each graph as @xmath315 . ] ancestral mean partial and specific distances , @xmath244 and @xmath247 , depending on mutation rates . the original hopfield fitness ( [ original hopfield fitness ] ) for three patterns has been used . results correspond to two typical examples of random , but correlated patterns chosen for sequences of lengths @xmath312 ( left ) , @xmath313 ( middle ) and @xmath314 ( right ) , specified at the top of each graph as @xmath315 . ] figure [ fig 6.21 ] shows some cases of the ancestral mean partial and specific distances , @xmath244 and @xmath247 , for the original hopfield fitness ( [ original hopfield fitness ] ) with three patterns , which are randomly chosen sequences of finite length . the correlations between the patterns ( and thus the variations of the @xmath279 ) are characteristic for the sequence length . the six cases of patterns shown here are typical examples for the sequence lengths considered . in the case of long sequences ( @xmath312 ) , the deviations of the patterns from the infinite sequence limit @xmath316 are small , and grow with decreasing sequence length . these correlations that are introduced into the system have the same effect as a choice of correlated patterns in the case of two patterns , such that the single critical mutation rate in the case of infinite sequence length is split up into two critical mutation rates , at each of which two of the @xmath209 show threshold behaviour . for short sequence length ( @xmath314 ) , it can be seen that , particularly at the smaller critical mutation rate , the threshold is smoothed out . ancestral mean partial and specific distances , @xmath244 and @xmath247 , depending on mutation rates . the hopfield - type fitness ( [ quadratic symmetric hopfield - type fitness ] ) with negative epistasis for three patterns and @xmath263 has been used . results correspond to the patterns used in figure [ fig 6.21 ] . ] ancestral mean partial and specific distances , @xmath244 and @xmath247 , depending on mutation rates . the hopfield - type fitness ( [ quadratic symmetric hopfield - type fitness ] ) with negative epistasis for three patterns and @xmath263 has been used . results correspond to the patterns used in figure [ fig 6.21 ] . ]    in figure [ fig 6.22 ] , the ancestral mean partial and specific distances , @xmath244 and @xmath247 , corresponding to the same patterns as in figure [ fig 6.21 ] are shown for the quadratic hopfield - type fitness ( [ quadratic symmetric hopfield - type fitness ] ) with negative epistasis and @xmath263 . for long sequence length , these look very similar to the results for infinite sequence length ( cf . figure [ fig 6.19 ] ) , showing clearly the single first order phase transition . for shorter sequence lengths , they become more and more smoothed out , such that at @xmath313 , roughly only every other pattern that was simulated shows an error threshold , whereas for @xmath314 , in the vast majority of cases , there is no threshold . note that this effect is present even though the finite sequence length was only simulated by choosing the patterns accordingly , so it is a feature of the model with correlated patterns .      in this section , the quadratic symmetric hopfield - type fitness given in equation ( [ quadratic symmetric hopfield - type fitness ] ) with negative epistasis ( i.e. , positive quadratic term ) was investigated for a small number of patterns . the results for the original hopfield fitness ( @xmath226 ) have been compared with those for the generalised quadratic hopfield - type fitness ( @xmath317 ) . furthermore , both uncorrelated patterns ( @xmath290 for all @xmath293 ) , corresponding to an infinite sequence length , and correlated patterns ( @xmath318 ) , simulating a finite sequence length , were considered . for two patterns , an analytical treatment was possible , making all values of the @xmath279 accessible , whereas the case of three or more patterns was treated numerically due to the larger number of variables . this means that apart from the uncorrelated choice of pattern ( @xmath290 ) , which was investigated for three , four and five patterns , only some correlated combinations for three patterns with @xmath319 were investigated , some typical examples of which are shown in section [ the case of three patterns ] . the results are summarised as follows :    * * original hopfield fitness ( @xmath226 ) * : * * for _ uncorrelated patterns _ , there is one second order error threshold for all @xmath209 at @xmath254 ( investigated @xmath320 ) . * * for _ correlated patterns _ , there are two second order error thresholds , each for half of the @xmath209 ( investigated @xmath321 ) . * * hopfield - type fitness with @xmath322 * : * * for _ uncorrelated patterns _ , there is a first order error threshold only on a restricted range of @xmath225 ( @xmath323 no first order threshold , @xmath324 $ ] ) . + for an even number of patterns ( @xmath325 ) , there is an additional second order error threshold for any @xmath326 ( @xmath327 , @xmath328 ) . this error threshold does not exist for the investigated cases of an odd number of patterns ( @xmath329 ) . * * for _ correlated patterns _ , at @xmath227 , there is one second order threshold , the other one , which is present for the original hopfield fitness , is smoothed out . at @xmath79 , there is up to one first order threshold , smoothed out for more strongly correlated patterns ( corresponding to shorter sequence length ) . the evaluation the hopfield - type fitness was limited to the cases of rather small numbers of patterns , simply because an increase in the number of patterns makes the evaluation more complex . however , the hopfield - type fitness was chosen as a potentially realistic fitness because of its ruggedness that can be tuned by the number of patterns chosen . the simple cases considered here probably do not show as high a degree of ruggedness as one would expect for realistic fitness functions . however , the results described here already indicate some features that are common for all numbers of patterns investigated here , and some that depend on whether the number of patterns is odd or even . it would be very interesting to establish whether these results generalise to an arbitrary number of patterns . furthermore , the concept of partitioning the set of sites into subsets , which was introduced to analyse the hopfield - type fitness , is very interesting . one could imagine a different interpretation for this by classifying sites according to the selection strength they evolve under . some of the behaviour identified for the hopfield system could also occur in such a setting : at intermediate mutation rates , partially ordered phases could exist , such that sites that evolve under weak selection have passed their error threshold and the population is in a phase that is disordered with respect to these sites , whereas at sites that are subject to strong selection the order is still maintained . the present work has been concerned with the investigation of a deterministic mutation  selection model in the sequence space approach , using a time - continuous formulation . important observables in these mutation selection models are the population and ancestral distributions @xmath23 and @xmath28 , and means with respect to these distributions , in particular the population mean fitness @xmath20 and the ancestral mean genotype @xmath206 . in equilibrium , @xmath23 is given by the right perron - frobenius eigenvector of the time - evolution operator @xmath144 , whereas the ancestral distribution is given by the product of both right and left pf eigenvectors @xmath23 and @xmath29 , @xmath30 . types have been modelled as two - state sequences . as mutation model , the single step mutation model was used , whereas selection was modelled by hopfield - type fitness functions , using the similarity of a sequence to a number of patterns to determine its fitness . this allows for a more rugged fitness landscape , and the complexity of the fitness can be tuned by the number of patterns . the large number of types that arise in the sequence space approach have been lumped into classes of types , labelled by the partial distances @xmath102 introduced in section [ lumping for the hopfield - type fitness ] as a generalisation of the hamming distance . with this , the maximum principle as developed by @xcite can be applied to the case of hopfield - type fitness functions as done in section [ the maximum principle ] , see also @xcite , treating two- and four - state sequences .    in section [ error thresholds ] , the maximum principle derived in section [ the maximum principle ] was used to investigate the phenomenon of the error threshold . these error thresholds can be detected with the maximum principle , because the delocalisation of the population distribution manifests itself as a jump ( or at least an infinite derivative with respect to the mutation rate ) of the ancestral mean genotype @xmath206 , the maximiser . not all fitness functions give rise to error thresholds , and as the error thresholds were first described for a model with highly unrealistic fitness function , it has been argued that they might be an artifact of this rather than a biologically relevant phenomenon . it is therefore clearly necessary to investigate more complex fitness functions with respect to this phenomenon .    here , quadratic hopfield - type fitness functions with small numbers of patterns have been investigated . for the original hopfield fitness , the results for all investigated numbers of patterns are identical . however , if the fitness differs from the original hopfield fitness , different behaviours are observed for different numbers of patterns . in the case of uncorrelated patterns , corresponding to random patterns chosen for infinite sequence length , the observed features seem to depend on whether the number of patterns is odd or even . because for correlated patterns , only the cases of two and three patterns were investigated , and found to behave differently , it would be interesting to see how these results generalise to a higher number of patterns .    in the original hopfield fitness , error thresholds were observed for all choices of patterns . this is not true for a generalised hopfield - type fitness . for instance , for a hopfield - type fitness with positive epistasis no thresholds were observed , going in line with the results for permutation - invariant fitness . but also for hopfield - type fitness functions with negative epistasis , there are not necessarily any thresholds , if the fitness deviates too much from the original hopfield fitness , challenging the commonly held notion that more complex fitness functions all tend to display error threshold behaviour . the complexity and ruggedness of the original hopfield fitness have been investigated @xcite and found to be good candidates for realistic fitness functions @xcite . however , these results do not necessarily transfer to the generalised hopfield - type fitness functions , and therefore it would be very useful to study these properties of a generalised hopfield - type fitness functions to analyse which of these factors are responsible for generating the thresholds . it is my pleasure to thank uwe grimm , michael baake , ellen baake and robert bialowons for helpful discussions and uwe grimm for comments on the manuscript . support from the british council and daad under the academic research collaboration programme , project no 1213 , is gratefully acknowledged . baake , e. , gabriel , w. , 2000 . biological evolution through mutation , selection , and drift : an introductory review . in : stauffer , d. ( ed . ) , annual reviews of computational physics vii . world scientific , singapore , pp . 203264 . holland , j.  j. , domingo , e. , de  la torre , j.  c. , steinhauer , d.  a. , 1990 . mutation frequencies at defined single codon sites in vesicular stromatitis - virus and poliovirus can be increased only slightly by chemical mutagenesis . journal of virology 64  ( 8) , 39603962 . loeb , l.  a. , essigmann , j.  m. , kazazi , f. , zhang , j. , rose , k.  d. , mullins , j.  i. , 1999 . lethal mutagenesis of with mutagenic nucleoside analogs . proceedings of the national academy of sciences of the usa 96 , 14921497 . sierra , s. , dvila , m. , lowenstein , p.  r. , domingo , e. , 2000 . response of foot - and - mouth disease virus to increased mutagenesis : of viral load and fitness in loss of infectivity . journal of virology 74  ( 18 ) , 83168323 .
a deterministic mutation  selection model in the sequence space approach is investigated . 
 genotypes are identified with two - letter sequences . 
 mutation is modelled as a markov process , fitness functions are of hopfield type , where the fitness of a sequence is determined by the hamming distances to a number of predefined patterns . using a maximum principle for the population mean fitness in equilibrium , the error threshold phenomenon is studied for quadratic hopfield - type fitness functions with small numbers of patterns . 
 different from previous investigations of the hopfield model , the system shows error threshold behaviour not for all fitness functions , but only for certain parameter values . 
 mutation 
 selection model , hopfield model , error threshold , maximum principle