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A New Monte Carlo Algorithm for Growing Compact Self Avoiding Walks

A New Monte Carlo Algorithm for Growing Compact Self Avoiding Walks
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    a  r   X   i  v  :  c  o  n   d  -  m  a   t   /   0   1   0   8   0   9   7  v   4   [  c  o  n   d  -  m  a   t .  s   t  a   t  -  m  e  c   h   ]   2   7   A  u  g   2   0   0   1 A new monte carlo algorithm for growing compact self avoiding walks S.L. Narasimhan 1 ∗ , P.S.R. Krishna 1 ∗ , K.P.N.Murthy 2+ and M. Ramanadham 1 † 1 Solid State Physics Division,Bhabha Atomic Research Centre, Mumbai - 400 085, India  2 Materials Science Division, Indira Gandhi Centre for Atomic Research,Kalpakkam - 603 102, Tamilnadu, India  We propose an algorithm based on local growth rules for kinetically generating self avoiding walkconfigurations at any given temperature. This algorithm, called the Interacting Growth Walk (IGW)algorithm, does not suffer from attrition on a square lattice at zero temperature, in cotrast to theexisting algorithms. More importantly, the IGW algorithm facilitates growing compact configura-tions at lower temperatures - a feature that makes it attractive for studying a variety of processessuch as the folding of proteins. We demonstrate that our algorithm correctly describes the collapsetransition of a homopolymer in two dimensions.36.20.Ey, 05.10.Ln, 87.10.+e The configurational properties of linear polymers un-dergoing a collapse transition at a tricritical tempera-ture T  θ , called the θ -point, have been studied extensivelybecause of their relevance to a wide variety of appli-cations such as, for example, the protein folding prob-lem [1]. The average radius of gyration (or equivalently,the average end-to-end distance) and the configurationalentropy of a long polymer chain have a universal  (i.e.,system-independent) behaviour characterized by the ex-ponents ν  and γ  respectively [2]. These exponents havedistinct sets of values for the three temperature regimes, T > T  θ , T  = T  θ and T < T  θ [2,3]. In order to under-stand the statistical nature of polymer conformations inthese three universal regimes, Interacting Self AvoidingWalk (ISAW) models with appropriate non-bonded near-est neighbour (nbNN) interactions have been proposed[4].Let S  N  denote an ensemble of equally weighted N  -stepSAW configurations, generated on a lattice by a stan-dard algorithm[5]. If  ǫ 0 is the energy associated with anynbNN contact, a SAW configuration with a total of  n NN  such contacts will have an energy E  = n NN  ǫ 0 . Hence,one can assign to it a Boltzmann weight proportionalto e − βE , where β  = 1 /k B T  , k B is the Boltzmann con-stant and T  the temperature. Such Boltzmann weightedSAW configurations constitute an ISAW ensemble, de-noted by I  N  ( β  ). By this definition, I  N  ( β  = 0) is thesame as S  N  because all the configurations of the formerhave the same probability of occurrence irrespective of their energies. Therefore, in the context of the ISAW en-semble, S  N  can be thought of as representing a polymerat ’infinite’ temperature. The statistical accuracy of anyphysical quantity averaged over I  N  ( β  ) becomes poorerat lower temperatures because significant contributioncomes from a smaller number of configurations [6]. Inorder to improve the statistics, especially at low temper-atures, it is necessary to generate as large an ensemble, S  N  , as possible; this process could become prohibitivelyslow due to severe attrition for large N  .A better solution is to devise an algorithm based onsuitable geometrical ( athermal  , or ’infinite’ temperature)rules for generating an ensemble, G N  , identically equiv-alent to I  N  ( β > 0). For example, the Kinetic GrowthWalk (KGW) [7] or the Smart Kinetic Walk (SKW) [8]on a hexagonal lattice straightaway generates an ensem-ble of configurations equivalent to the ISAW ensemble,  I  N  ( β  = ln 2). Having generated the athermal  ensemble, G N  , by such a geometric algorithm, ensemble averagescorresponding to a lower temperature could be obtainedby Botzmann weighting these configurations appropri-ately. This would ensure better statistical accuracy ascompared to what could be obtained directly from S  N  .Yet, whether it is possible at all to sample a statisticallysignificant number of maximally compact configurationsis a moot point to consider because it involves a ’zero’temperature sampling.In this paper, we present an algorithm for kineticallygrowing a SAW configuration at any given temperature T  ≥ 0. This algorithm, called the Interacting GrowthWalk (IGW) algorithm, is able to generate more accuratedata for longer walks at lower temperatures because sam-ple attrition is less severe at lower temperatures. In fact,on a square lattice, the walk grows indefinitely into max-imally compact configurations at T  = 0, in contrast tothe conventional sampling algorithms [9, 10]. We demon-strate that our algorithm is capable of describing theuniversal behaviour of a SAW above, at and below the θ -point in two dimensions.We also present a speculativeFlory thory for the IGW.We start the growth process by ’occupying’ an arbi-trarily chosen site, r 0 , of a regular d -dimensional latticeof coordination number z whose sites are initially ’unoc-cupied’ (by monomers). The first step of the walk canbe made in one of the z available directions, by choosingan ’unoccupied’ NN of  r 0 , say r 1 , at random. Let thewalk be non-reversing so that it has a maximum of  z − 11  directions to choose from for any further step made. Let { r mj | m = 1 , 2 ,...,z j } be the ’unoccupied’ NNs availablefor the j th step of the walk. If  z j = 0, the walk cannot grow further because it is geometrically ’trapped’. Itis, therefore, discarded and a fresh walk is started from r 0 . If  z j  = 0, the walk proceeds by choosing one of theavailable sites with a probability defined as follows:Let n mNN  (  j ) be the number of nbNN sites of  r mj . Then,the probability that this site is chosen for the j th step isgiven by,  p m ( r j ) ≡ exp [ − βn mNN  (  j ) ǫ 0 ] z j  m =1 exp [ − βn mNN  (  j ) ǫ 0 ](1)where the summation is over all the z j available sites. At’infinite’ temperature ( β  = 0), the local growth probabil-ity, p m ( r j ), is equal to 1 /z j and thus the walk generatedwill be the same as the KGW. However, at finite tem-peratures, the walk will prefer to step into a site withmore or less nbNN contacts depending on whether ǫ isnegative or positive. The probability of kinetically gener-ating a walk configuration, C ≡ { r 0 , r 1 ,..., r j ,... } , is thengiven by P  C =  j p ( r j ). We set ǫ 0 equal to − 1 with-out loss of generality so that β  could correspond to thedimensionless temperature. (a)   (b)(c)(d)(e)(f) FIG. 1. Typical configurations of a 1000-step walk on asquare lattice for β  = 0( a ) , 2 . 0( b ) , 3 . 0( c ) , 4 . 0( d ) , 5 . 0( e ) and300( f  ). In Fig.1, we have shown the typical configurationsof a 1000-step walk on a square lattice for β  =0 , 2 . 0 , 3 . 0 , 4 . 0 , 5 . 0 and 300. Evidently, the walk growsinto a more compact configuration at lower temperatures,made up of a chain of square blobs having ’helical’ and’sheetlike’ structures.We have generated ten million configurations of walksupto 2500 steps for various values of  β  , and obtainedthe mean square end-to-end distance, < r 2 ( N  ) > , as asimple unweighted average ( i.e., < r 2 ( N  ) > =  C r 2 /  N  ,where the summation is over all the N  configurationsgenerated). We have presented < r 2 ( N  ) > as a functionof  N  in Fig.2. 101001000110100100010000    l   n  <   L  >   β Slope = 1.0Slope = 1.5   <   r    2    (   N   )  > N 0 1 2 34567 FIG. 2. Log-log plot of the mean square end-to-end dis-tance as a function of  N  for β  = 0 , 1 . 0 , 1 . 5 , 2 . 0 , 2 . 5 , 3 . 0 , 4 . 0 , 5 . 0and 300, from top to bottom. Inset: Logarithm of the meantrapping length, ln < L > as a function of  β    10 100 10000.500.75   <   r    2 (   N   )  >    1   /   2 /   N    4   /   7 N FIG. 3. Semi-log plot of  < r 2 ( N  ) > 1 / 2 /N  4 / 7 as a functionof log(N) for β  = 3 . 0 , 3 . 5 , 3 . 75 , 3 . 9 , 4 . 0 , 4 . 25 , 5 . 0 and 300, fromtop to bottom Sample attrition is the most severe problem for β  = 0and it becomes less and less severe as the value of  β  increases. Consequently, we have presented the data uptoa maximum of  N  = 350 for β  = 0 and N  = 2500 for β  = 300. It is clear that the dotted lines with slopes 1 . 5and 1 . 0 indicate the asymptotic behaviour of the datafor β  = 0 and β  → ∞ , corresponding to the SAW andthe collapsed walk limits respectively. We do not know a priori  whether a collapse transition exists for our walk.We assume that it exists and is in the same universality2  class as the θ -point, and then check if our data supportthis assumption.Since it is known that the exponents, ν  and γ  , havethe exact values 4 / 7 and 8 / 7 at θ -point in two dimen-sions [4], we have plotted < r 2 ( N  ) > 1 / 2 /N  4 / 7 as afunction of  log ( N  ) in Fig.3. The data tend to flat-ten out for β  ∼ 4 implying thereby that the θ -pointis located near this value of  β  . We have also plot-ted < r 2 ( N  ) > /N  8 / 7 as a function of  β  in Fig.4 for N  = 800 , 1000 , 1200 , 1400 , 1600 , 1800 and 2000. Thecrossover value of  β  ( ∼ 4 in our case) is expected [11]to correspond to the θ -point value. 2.0 2.5 3.0 3.5 4.0 4.5   <   r    2 (   N   )  >   /   N    8   /   7   β FIG. 4. < r 2 ( N  ) > /N  8 / 7 as a function of  β  for N  = 800to 2000 in steps of 200 from bottom to top. Independently, we have obtained the exponent γ  fromthe fraction of successful walks, S  ( N  ) ∼ N  γ − 1 e − λN  ,where λ is the attrition constant and plotted them forsix different values of  β  in Fig.5. We find that γ  has avalue ( ∼ 1 . 13) close to the expected theoretical value 8 / 7for β  = 4.Further evidence that it is indeed close to the θ -pointis presented in Fig.6, where we have plotted the crossoverexponent, φ ( N  ), as a function of 1 /N  at β  = 4 using theprescription of Grassberger and Hegger [12]. The solidline is a quartic polynomial fit to the data drawn so asto guide the eye. The extrapolated value (0 . 419 ± 0 . 003)for φ is close to the expected exact value 3 / 7. 3.0 3.5 4.0 4.5 θ - point        γ β FIG. 5. The exponent γ  as a function of  β  . Correspondingto the θ -point, γ  has a value ∼ 1 . 13. 0.000 0.001 0.002 0.003 0.004 0.0050.   β = 4.0            φ    (   N   ) 1/N FIG. 6. The crossover exponent, φ , as a function of 1 /N  .The solid line is a quartic polynomial fit and is drawn to guidethe eye. The extrapolated value is ∼ 0 . 419 ± 0 . 003 All these figures put together suggest that a collapsetransition for this walk exists and the corresponding di-mensionless nbNN contact energy is close to − 4.The walk configuration, C , having a total of  n NN  ( C ) =  N j =1 n NN  (  j ) non-bonded NN contacts, is grown withthe probability, P  C = exp [ − n NN  ( C ) βǫ 0 ]  N j =1  z j  m =1 exp [ − n mNN  (  j ) βǫ 0 ]  (2)It is possible to write the denominator, W  ( C ), of theabove equation as e − n NN ( C ) β ′′ ǫ 0 , where β  ′′ is an effec-tive inverse temperature. The value of  β  ′′ will be less(greater) than that of  β  if  ǫ 0 is positive (negative or zero).Nevertheless, ISAW algorithm can not sample the walkat an effective temperature given by β  ′ ≡| ( β  ′′ − β  ) | be-cause β  ′′ can only be estimated a posteriori  on the basisof the configuration generated. An alternative is to havea kinetic algorithm, such as what we have proposed inthis paper, which grows a walk by sampling the availablegrowth sites as per their local  energies. This is in con-trast with the ISAW algorithm which samples fully grownand equally weighted SAW configurations ( i.e., chains )according to their total  energies. To underline this basicdifference, we refer to our walk as the Interacting GrowthWalk (IGW).It is appropriate at this juncture to note that the dif-ference between our algorithm and the PERM algorithm(method B) of Grassberger [10] is analogous to that be-tween the Rosenbluth-Rosenbluth algorithm (RR) [13]and the KGW [7]. Ours is the finite temperature gener-alisation of the KGW, just as PERM is the finite tem-perature generalisation of the RR method. There is no a priori  reason therefore to expect that IGW will belong tothe same universality class as ISAW, they both being dif-ferent models altogether. Yet, our data seem to suggestthat it may well be so.Since the IGW is equivalent to the KGW in the limit β  → 0, it is of interest to see if survival probability argue-3  ments a la  Pietronero [14] could be devised for describ-ing its asymptotic behaviour even if only tentatively. Let T   N  be an ensemble of  N  -step True Self Avoiding Walk(TSAW) [15] configurations whose end-to-end distancesare known to be Gaussian distributed in a space of di-mension d ≥ 2. As we move along an arbitrarily chosenconfiguration, we try to estimate the probability of sur-viving self-intersections and geometrical trappings. Thisinvolves accounting for the probability per step of en-counter, p E , and the probability of trapping, p T  whichtogether determine the survival of the walk. Assumingthat the trapping probability per step is a constant andalso that the encounter probability per step, p E ∼ ρ αN  ,where ρ N  is the chain density and α is the order of en-counter ( i.e., the number of nbNN contacts), it has beenshown that ν  = ( α + 2) / ( dα + 2) for the KGW.The observed fact that the IGW becomes more com-pact at lower temperatures (Fig.1) implies, within theframework of the above Flory-like arguments, that thereshould be an enhancement, q E [ ρ N  ], of the encounterprobability per step, p E . We expect q E [ ρ N  ] to increaseimplicitly as a function of  β  subject to the condition that q E [ ρ N  ] → 1 as β  → 0. On the other hand, since the meantrapping length of IGW has been found to increase expo-nentially with β  (inset of Fig.2), the trapping probabilityper step may be expected to be attenuated by a factorproportional to exp ( − β  ). So, if we assume an implicittemperature dependence, q E [ ρ N  ] ∼ ρ βN  , we can show that ν  = ( α + β  +2) / [ d ( α + β  )+2]. While it obviously reducesto the Pietronero’s formula in the limit β  → 0, it reducesto the form ν  = 1 /d for the collapsed state in the limit β  → ∞ . Since first order encounter ( α = 1) is sufficientto trap the walk, we have ν  = ( β  + 3) / 2( β  + 2) in twodimensions. This yields the value, β  θ = 5, correspondingto the exact θ -point exponent ν  = 4 / 7. It may be notedthat this value is fortuitously close to our numericallyestimated value. However, in order to ensure universal-ity of  ν  , we should have a term proportional to the ratio β/β  θ (say,˜ β  ≡ Kβ/β  θ ) rather than β  itself in the for-mula. The proportionality constant K  may then be fixedby the θ -point value of  ν  : K  + α = 2(1 − ν  θ ) / ( dν  θ − 1), d = 1 being a pathological case. The fact that the first-order encounter does not trap the walk at T  = 0 impliesthat α also has some temperature dependence. Moreover,the continuous dependence of  ν  on β  which the above for-mula suggests is at variance with the fact that there areonly three universal regimes corresponding to β <, = and > β  θ respectively. This needs further study.We thus have a powerful growth algorithm for generat-ing SAW configurations at any given temperature, T  ≥ 0.Its strength lies in the fact that it suffers less attritionand is able to selectively grow compact configurations atlower temperatures. Because it is capable of generatingmaximally compact configurations at zero temperature,it may prove to be a very useful algorithm for studyingprotein folding processes. We have also demonstratedexplicitly in two dimensions that it correctly describesthe collapse transition of a homopolymer. Whether it isexactly the same as the (ISAW) θ -point is an interest-ing open question, especially because the minimum walklength required to be in the asymptotic regime increasesexponentially with the inverse of temperature even in twodimensions.We are thankful to T. Prellberg and S. Bhattacharjeefor helpful comments on this work. ∗; +; † [1] D. Napper, Polymeric Stabilization of Colloidal Disper-sions (Academic, NY, 1983); H. S. Chan and K. Dill,Annu. Rev. Biophys. Biophys. Chem. 20 , 447 (1991);C. Vanderzande, Lattice models of polymers (CambridgeUniv. Press, UK, 1998); T. Prellberg, Lattice models of interacting polymers and vesicles (Habilitation Thesis,Technische Universit¨ a t Clausthel, April 2001).[2] P.G. de Gennes, Scaling concepts in polymer physics (Cornell Univ. Press, 1979).[3] K. Barat and B. K. Chakrabarti, Phys. Rep. 258 , 377(1995).[4] H. Saleur, J. Stat. Phys. 45 , 419 (1986); B. Duplantierand H. Saleur, Phys. Rev. Lett. 59 , 539 (1987).[5] A.D. Sokal, preprint hep-lat  /9405016.[6] P.H. Poole, A. Coniglio, N. Jan and H.E. Stanley, Phys.Rev. B39 , 495 (1989); A. Coniglio, N. Jan, I. Majid andH.E. Stanley, Phys. Rev. B35 , 3617 (1987).[7] N. Majid, N. Jan, A. Coniglio and H.E. Stanley, Phys.Rev. Lett. 52 , 1257 (1984); K. Kremer and J.W. Lyk-lema, Phys. Rev. Lett. 55 , 2091 (1985); N. Majid, N.Jan, A. Coniglio and H.E. Stanley, Phys. Rev. Lett. 55 ,2092 (1985).[8] A. Weinrib and S.A. Trugman, Phys. Rev. B31 , 2993(1985).[9] F. Seno and A. L. Stella, J. Phys. (Paris), 49 , 739 (1988).[10] P. Grassberger, Phys. Rev. E56 , 3682 (1997).[11] M.P. Taylor and J.E.G. Lipson, J. Chem. Phys. 109 , 7583(1998).[12] P. Grassberger and R. Hegger, J. Phys. (Paris) I5 , 597(1995).[13] M.N. Rosenbluth and A.W. Rosenbluth, J. Chem. Phys. 23 , 356 (1955).[14] L. Pietronero, Phys. Rev. Lett. 55 , 2025 (1985); A.L.Stella, Phys. Rev. Lett. 56 , 2430 (1986); L. Pietronero,Phys. Rev. Lett. 56 , 2431 (1986).[15] D.J. Amit, G. Parisi and L. Peliti, Phys. Rev. B27 , 1635(1983); L. Pietronero, Phys. Rev. B27 , 5887 (1983); J.Bernasconi and L. Pietronero, Phys. Rev. B29 , 5196(1984). 4
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