An ensemble of directed macromolecules on a lattice is considered, where the
constituting molecules are chosen as a random sequence of N different types.
The same type of molecules experiences a hard-core (exclusion) interaction. We
study the robustness of the macromolecules with respect to breaking and
substituting individual molecules, using a 1/N expansion. The properties depend
strongly on the density of macromolecules. In particular, the macromolecules
are robust against breaking and substituting at high densities.Comment: 9 pages, 4 figure

Banavar J R

Berezin F A

Ch Moseley

de Gennes P G

K Ziegler

Li Y-Q

English

arXiv

Moseley, Christopher

Ziegler, Klaus

OPUS - Augsburg University Publication Server

Correlations in systems of complex directed macromolecules

Ziegler, Klaus G.

OPUS Augsburg

                                                                                                                                    Correlations in systems of complex directedmacromoleculesCh Moseley and K ZieglerInstitut für Physik, Universität Augsburg, Germany                                                                                                     AbstractAn ensemble of directed macromolecules on a lattice is considered, wherethe constituting molecules are chosen randomly with N different colours.Molecules of the same colour experience a hard-core (exclusion) interaction.We study the robustness of the macromolecules with respect to breaking andchanging the colour of a constituting molecule, using a 1/N expansion. Theproperties depend strongly on the density of macromolecules. In particular, themacromolecules are robust against breaking and changing the colour at highdensities but fragile at intermediate densities.                                                              1. IntroductionIn a liquid of molecules long chain-like macromolecules (MM) can be formed due to a specialsteric intermolecular interaction. These MM are often called polymers [1]. We assume thatthe MM are directed, i.e. they have a preferred direction in the d-dimensional space. Thisimplies that they are stretched and do not form loops or overhangs.In many systems the constituting molecules (CM) of the MM are identical. However,in more complex systems, like in biological substances, the CM are chemically different.An example for the latter are the sequences of amino acids in proteins [2–4] and molecularsequences along a DNA helix. For our statistical model we assume that there are N differentkinds of molecules, represented by N colours. They are randomly chosen to contribute tothe MM such that the resulting MM can be viewed as a chain of coloured beads (cf figure 1).An ensemble of MM can be considered as being created in a liquid of molecules by a stericinteraction among the CM. As a result, we have a space-filling system of molecules, wheresome are constituting the MM and others exist as individual molecules. Properties of theensemble on large scales should not depend on local structures. This allows us to consider alattice model [3], where each lattice site is occupied by a molecule of each type. It is assumedthat molecules of the same type have a strong repulsive interaction. This restricts us to at mostone molecule of each type at any given lattice site. The fact that the interaction of differentmolecule types is neglected can be understood as a chemical property of the molecules.                                                               S1809S1810                     Figure 1. Two directed macromolecules.Any pair of neighbouring molecules can form a bond state, representing a pair ofmolecules, which acts as a building block for an MM. There is a tendency of these bondstates to bond with other molecules. This leads eventually to an equilibrium state, where wehave a mixture of only large MM and individual molecules. The repulsive interaction betweenthe molecules creates correlations among the MM which lead to interesting properties of theensemble. For instance, the density of MM as a function of the probability of a local bondstate is of interest, as well as the question of how likely it is to break an MM or to replace aCM of special type by one of a different type. The latter can happen during a collision processwhen an individual molecule collides with an MM and exchanges with a CM of the MM.We consider an equilibrium ensemble of lattice MM and define three quantities to describeits global properties with respect to the molecular structure. There are two quantities thatmeasure the correlations in real space. One measures how likely it is to break a directed MMand take it apart in space (cf figure 2); the other one measures how likely it is to create a new(finite) MM by changing the type of the CM in an MM. If the ends of the broken MM or theends of the new (finite) MM are x and x ′ we write for the corresponding quantities Gxx′ andHxx′ , respectively. It will be shown that Gxx′ and Hxx′ are directly related. Another quantity isthe coherence in α (molecular-component) space. This is called Sαα′ and measures how likelyit is to replace a CM of type α in an MM by one of type α′ (cf figure 3). All these quantitiesare evaluated in the limit N → ∞ and in a 1/N expansion. We apply a functional-integralformalism developed in [5] to perform the calculations in a straightforward manner.2. The modelA statistical model for the equilibrium statistics of directed MM was introduced in [5]. Webriefly summarize the construction of the model and then turn to the evaluation of the interestingcorrelations mentioned in the introduction. The model is based on the idea that on each site ofa (cubic) lattice N different molecules are located. They exist either as individual (separated)molecules of a liquid with relative weight µ or contribute to a directed MM as a bond state ofa nearest-neighbour molecular pair at x, x ′ of types α and α′. Moreover, the MM are directed,i.e. there is a special direction t in the d-dimensional space as a preferred direction of the MM.The fluctuations of the MM are perpendicular to this, indicated by the (d − 1)-dimensionalvector r . Therefore, we write x = (t, r) for a site on the lattice. The relative weight of aS1811Figure 2. Breaking of a macromolecule.Figure 3. Change of the type of a constituting molecule inside a macromolecule, caused by acollision with a molecule from the surrounding liquid.nearest-neighbour molecular pair at x, x ′ of type α and α′ iswαα′x,x′ =1Nwx,x′ ={J/N if t ′ = t + 1, r = r ′ or r, r ′ nearest neighbours0 otherwise.The directed chain-like structure is given by the matrix w that is asymmetric in the direction tbut symmetric with respect to r . x ′ is assumed to be ‘later’ along the direction of the MM. wis also symmetric with respect to α, α′. Thus α and r are of the same type in contrast to t .The statistical distribution of configurations of MM has to take into account a strongrepulsive interaction between molecules of the same type at the same site. This can convenientlybe described by a commutative algebra of nilpotent variables ηα,σx (i.e., (ηα,σx )l = 0 if l > 1)with α = 1, 2, . . . , N and σ = 1, 2 [5]. Each molecule, characterized by its position x = (r, t)on the lattice and by its chemical character (α, σ ), is described by a variable. Since {ηα,σx } arenilpotent, the repulsive interaction results in an exclusion principle for molecules of the sametype. The atoms are considered as classical objects. Consequently, products of ηα,σx must beS1812                    commutative. A segment of the MM (i.e. a bond state of two molecules or a dimer) readswαα′x,x′ ηα,1x ηα′,2x′ .A linear mapping of the algebra {ηα,σx } to the complex numbers can be written as an integralon a lattice  with [6]∫ ∏x∈′⊆∏α∈ix∏σ∈ jx,αηα,σx ={1 if ′ = , ix = 1, 2, . . . , N , jx,α = 1, 20 otherwise.The integral vanishes if the product is incomplete with respect to the lattice , to α =1, 2, . . . , N or to σ = 1, 2. It can be used to write the statistical weight PI of a configurationI of MM asPI = 1Z∫WIwithWI =∏(x,α;x′ ,α′)∈I1Nwx,x′ ηα,1x ηα′,2x′∏(x,α)(1 + µηα,1x ηα,2x ),where (x, α; x ′, α′) is a bond between two CM of type α and α′ at sites x and x ′. Thenormalization Z is the sum over all possible configurations {I }:Z =∑{I }∫WI .Finally, we impose periodic boundary conditions on the lattice. This implies a system on atorus with closed loops of MM. However, the choice of boundary conditions should not becrucial for the local properties of the MM like the density of the MM, Gxx′ , Hxx′ or Sαα′ .3. Functional-integral representationIntroducing analytic functions of the nilpotent field and using the properties of the integral, wemay write for the normalizationZ =∑{I }∫WI =∫exp[∑x,x′N∑α,α′=1(1Nwx,x′ + µδx,x′ δα,α′)ηα,1x ηα′,2x′]. (1)In other words, we can replace under the integral∑{I }WI → W = exp[∑x,x′N∑α,α′=1(1Nwx,x′ + µδx,x′ δα,α′)ηα,1x ηα′,2x′]in Z . Another important consequence of the properties of the nilpotent field is that we cancalculate the probability that a point x, α belongs to a MM configuration I :1Z∫(1 − µηα,1x ηα,2x )WI ={PI if (x, α) ∈ I0 otherwise.After summation over all configurations {I }, the probability of a molecule of type α at site xbeing part of any MM is∑{I }PI =∑{I }1Z∫(1 − µηα,1x ηα,2x )WI =1Z∫(1 − µηα,1x ηα,2x )W.                                                    S1813This is also the local density n(µ)of an MM. In a similar manner, we can express the correlationsin terms of normalized integrals〈· · ·〉 = 1Z∫· · · Was correlations in real spaceGxx′ = 〈ηα,1x ηα,2x′ 〉 = −C(xα; x ′α),Hxx′ = 〈ηα,2x ηα,1x′ 〉 = −C(x ′α; xα),and correlations in α spaceSαα′ = 〈ηα,1x ηα′,2x 〉 = −C(xα; xα′).These are correlation functions in real space and in α space, respectively. The density of MMcan also be expressed asn(µ) = 1 − µC(xα; xα).A lengthy but straightforward calculation [5] shows that the normalization Z in equation (1)can be written as an functional integral with respect to complex fields ϕ and χ asZ =∫e−N S∏xdϕx dχx . (2)withS = (ϕ, (1 + w)−1ϕ̄) + (χ, χ̄) −∑xlog[µ + (ϕx + iχx)(ϕ̄x + iχ̄x )].An analogous calculation for the correlation functions leads to〈· · ·〉 = 1Z∫· · · e−N S∏xdϕx dχx . (3)The correlation functions then becomeC(xα; xα) =〈[µ + (ϕx + iχx)(ϕ∗x + iχ∗x )]−1〉C(xα; x ′β) = µ〈(ϕx + iχx)(ϕ∗x′ + iχ∗x′ )[µ + (ϕx + iχx)(ϕ∗x + iχ∗x )][µ + (ϕx′ + iχx′ )(ϕ∗x′ + iχ∗x′ )]〉for (x, α) = (x ′, β).3.1. 1/N expansionThe integrals in equations (2) and (3) depend on the number of different molecule types Nonly through the prefactor N in the exponential. This enables us to perform a saddle-pointintegration for large values of N [5]. The result is a 1/N expansion. The homogeneous saddlepoint of the integral in equation (2) satisfies the relationiχ = − 12ϕ, iχ̄ = − 12 ϕ̄.Then there is a trivial solution ϕ0 = ϕ̄0 = χ0 = χ̄0 = 0 and a non-trivial solution withϕ1ϕ̄1 = 4(1 − µ).An expansion of the exponent in Z in terms of the deviations around the saddle pointφx = ϕ + iχ + δφx with δφx := δϕx + iδχx , δφ∗x := δϕ∗x + iδχ∗xS1814                     provides corrections in powers of 1/N . For a non-zero density of MM (i.e. ζ ≡ µ−1 > 1), anexpansion up to second order leads toC(xα; x ′β) ≈ ζ−1 − ζ−2 + 1ζ 3[−(ζ − 1)〈δφ2x 〉 − 2(ζ − 1)〈δφxδφ∗x 〉− (ζ − 1)〈δφxδφx′ 〉 + 〈δφxδφ∗x′ 〉 +√ζ(ζ − 1)〈δφx 〉 + (ζ − 1)2〈(δφ∗x )2〉+ (ζ − 1)2〈δφ∗x δφx′ 〉 − (ζ − 1)〈δφ∗x δφ∗x′ 〉− (ζ − 1)√ζ(ζ − 1)〈δφx 〉 + (ζ − 1)2〈δφ2x′ 〉 − 2(ζ − 1)〈δφx′δφ∗x′ 〉− (ζ − 1)√ζ(ζ − 1)〈δφx′ 〉 − (ζ − 1)〈(δφ∗x′ )2〉 + √ζ(ζ − 1)〈δφ∗x′ 〉].The first term is the large-N limit; all other terms either vanish because of 〈δφx 〉 = 0 or are oforder 1/N because of 〈δφ2x′ 〉 = O(1/N). The saddle-point calculation leads to the result thatthe first-order terms vanish, and the second-order terms can be expressed as the integrals(r − r ′, t − t ′) := 〈δφxδφx′ 〉 =〈δφ∗xδφ∗x′〉≈ 1N∫ ∫ 2π0B(k)2(ζ − 1)2B(k) cos(ω) − ζ + (ζ − 2)B(k)2 ei[k(r−r ′ )−ω(t−t ′)] dω2πdd−1k(2π)d−1(4)̂(r − r ′, t − t ′) := 〈δφxδφ∗x′ 〉≈ 1N∫ ∫ 2π0B(k)(B(k) − ζ cos(ω))2B(k) cos(ω) − ζ + (ζ − 2)B(k)2 ei[k(r−r ′ )−ω(t−t ′)] dω2πdd−1k(2π)d−1(5)with the function B(k) := 1 − J + Jd−1∑d−1j=1 cos k j .4. Discussion of the resultsThe density of MM isn(µ) ={1 − µ + O(1/N) for µ < 10 for µ > 1.It increases with decreasing N , as shown in figure 4. This means that it is more likely to forma MM if the number of available kinds of molecules is reduced.How likely it is to break a MM depends on the position of the remaining ends x − x ′.The corresponding quantity Gxx′ decays with |x − x ′| which indicates that the molecules havea tendency to form a MM. However, it does not vanish asymptotically but stays nonzero forζ > 1 and t ′ = t:G = lim|r−r ′|→∞ C(xα; x′α) ≈ (1 − ζ−1)(ζ−1 +[− 4ζ 2̂(0, 0) +2ζ(1 − 2ζ)(0, 0)]).The ζ dependence of this asymptotic result is shown in figure 4. The same property is validfor the creation of a finite chain of molecules with end points x, x ′, described by Hxx′ .The correlation in α space Sαα′ , describing how likely it is to change the type of moleculesin the MM, does not depend on the specific pair α, α′ as long as α = α′ because the modelconnects all α = α′ with equal probability:S = C(xα; xα′) ≈ ζ−1 − ζ−2 + 1ζ(1 − 6ζ+6ζ 2)̂(0, 0) +2ζ(1 − 1ζ)2(0, 0).                                                    S18150.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0µmolecular correlation S (N=5)spacial correlation G and H (N=5)correlation function (N=infinity)particle density (N=5)particle density (N=infinity)Figure 4. The density of macromolecules and the correlations for breaking a macromolecule (G)and replacing a molecule in a macromolecule (S).The N → ∞ terms of both correlations G and S agree but there is a difference in terms oforder 1/N (see figure 4). It is unlikely to break an MM and separate the pieces at low and athigh density. This is a consequence of the fact that there are either too few MM or too manyat high densities such that the separation of the pieces is difficult. It is remarkable that alreadyat densities less than 1 the separation is completely blocked, as indicated by the vanishing Gfor nc  n  1. Moreover, the replacement of a CM is most likely at intermediate densitiesbut suppressed at low and high densities. Again, there is a critical density n′c < 1 such thatthe substitution is blocked for n′c  n  1. This can be understood as an effect of interaction:since there is a exclusion between molecules of the same type, we can only substitute if thereis no contribution of this molecule from another MM. At high densities it is very unlikely thata site is not occupied yet by a CM of a specific type from another MM.4.1. Algebraic decay of the correlation functionTo analyse asymptotic behaviour of the correlation function for large distances |r − r ′| of athree-dimensional system, we first have to perform the ω-integration in equations (4) and (5).This leads to(r − r ′) = 1N∫ −B(k)(ζ − 1)√((ζ − 2)B(k)2 − ζ )2 − 4B(k)2 eik(r−r ′) d2k(2π)2,̂(r − r ′) = 1N∫ −ζ2+ζ 22 + B(k)2(−1 + ζ − ζ 22)√((ζ − 2)B(k)2 − ζ )2 − 4B(k)2eik(r−r ′) d2k(2π)2.S1816                     The spatially decaying part of the correlation function is given as2ζ 2(1 − 1ζ)(r − r ′) + 1ζ(1 − 2ζ+2ζ 2)̂(r − r ′).For small values of k we can assume B(k) ≈ 1 in the numerators of (r − r ′) and ̂(r − r ′).The square root in the denominators can be approximated by using B(k) = 1 − k22 + O(k4):√((ζ − 2)B(k)2 − ζ )2 − 4B(k)2 =√4(ζ − 1)k2 + O(k4) ≈ 2√ζ − 1|k|.Therefore, the asymptotic behaviour of the correlation function is proportional toconst∫eik·(r−r ′)|k|d2k(2π)2∝ const∫ 2π0∫ ∞0eik|r−r′ | cos φ dk dφ= |r − r ′|−1 const∫ 2π0∫ ∞0eiκ cos φ dκ dφ︸ ︷︷ ︸= const∝ |r − r ′|−1,where k has been replaced by κ = k|r − r ′|. Thus the spatial correlations decay like 1/|r − r ′|for d = 3.5. ConclusionsA statistical ensemble of directed MM, constructed with a random sequence of N differentmolecules, has been treated for large values of N in terms of a 1/N expansion. We havestudied the robustness of the MM with respect to breaking and replacement of individual CM.Breaking a MM and taking the pieces apart in space reveals a long-range correlation with analgebraic decay with the inverse distance. The properties in space and in α are identical in theN → ∞ limit but different in 1/N .At high densities the broken MM cannot be separated to arbitrarily large distances dueto the interaction between different MM. The replacement of individual CM is most likely atintermediate densities but strongly suppressed at high densities. This is also a consequence ofthe repulsive interaction between the molecules of the same type.References[1] de Gennes P G 1979 Scaling Concepts in Polymer Physics (Ithaca, NY: Cornell University Press)[2] Li Y-Q et al 2004 Preprint q-bio.BM/0408024[3] Banavar J R and Maritan A 2003 Rev. Mod. Phys. 75 23[4] Banavar J R 2004 Preprint q-bio.BM/0410031[5] Ziegler K 1991 Physica A 179 301[6] Berezin F A 1966 Method of Second Quantization (New York: Academic)

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