A semiflexible polymer chain under geometrical restrictions: Only bulk behaviour and no surface adsorption

We analyse the conformational behaviour of a linear semiflexible homo-polymer chain confined by two geometrical constraints under a good solvent condition in two dimensions. The constraints are stair shaped impenetrable surfaces. The impenetrable surfaces are lines in a two dimensional space. The infinitely long polymer chain is confined in between such two (A and B) surfaces. A lattice model of a fully directed self-avoiding walk is used to calculate the exact expression of the partition function, when the chain has attractive interaction with one or both the constraints. It has been found that under the proposed model, the chain shows only a bulk behaviour. In other words, there is no possibility of adsorption of the chain due to restrictions imposed on the walks of the chain.


Introduction
Biopolymers (DN A & proteins) are soft object and therefore such molecule can be easily squeezed into the spaces that are much smaller than natural size of the molecule. For example, actin filaments in eukaryotic cell or protein encapsulated in Ecoli [1] are the examples of confined molecules that serves as the basis for understanding molecular processes occurring in the living cells. The conformational properties of single biomolecule have attracted considerable attention in recent years due to developments in the single molecule based experiments [2,3]. Under confined geometrical condition, excluded volume effect and effect of geometrical constraint compete with entropy of the molecule. Therefore, geometrical constraints can modify conformational properties and adsorption desorption transition behaviour of the molecules. 1 The behaviour of a linear flexible polymer molecule under good solvent condition, confined to different geometries have been studied for past few years [4,5,6] and semiflexible molecule based theoretical studies under confined geometry also find considerable attention in recent years [7,8] and also see, references quoted therein. For example, Whittington and his coworkers [4,5,6] used directed self avoiding walk model to study behaviour of a flexible polymer chain confined between two parallel walls on a square lattice and calculated force diagram for surface interacting polymer chain. Rensburg et. al [6] showed through numerical studies using isotropic self avoiding walk model that force diagram obtained for surface interacting polymer chain confined in between two parallel plates have qualitatively similar phase diagram to that has been obtained by Brak et. al [4] for directed self avoiding walk model.
However, in present investigation, we have considered a linear semiflexible polymer chain confined in between two one dimensional stair shaped impenetrable surface (geometrical constraint) under good solvent condition and we have discussed adsorption phase transition behaviour of an infinitely long linear semiflexible polymer chain on the constraint(s). Such investigation may be useful to understand biological process of macromolecule and membrane interactions as well as such process of biotechnology as DN A in micro-arrays and electrophoresis.
To analyze adsorption phase transition behaviour of the semiflexible chain, confined in between the constraints, we have chosen fully directed self avoiding walk model introduced by Privmann and his coworkers [9] and used generating function technique to solve the model analytically for different values of spacing between the constraints. The results so obtained is used to discuss adsorption phase transition behaviour of the polymer chain on the stair shaped surface and also to compare the results obtained for adsorption transition of the polymer chain on a flat surface [10,11].
Since, the constraint is an attractive surface, therefore,it contributes an energy ǫ s (< 0) for each step of the fully directed self avoiding walk making on the constraint. This leads to an increased probability defined by a Boltzmann weight ω = exp(−ǫ s /k B T ) of making a step on the constraint (ǫ s < 0 or ω > 1, T is temperature and k B is the Boltzmann constant). The polymer chain gets adsorbed on the constraint at an appropriate value of ω or ǫ s . Therefore, transition between adsorbed to desorbed phase is marked by a critical value of adsorption energy ǫ s or ω c . The crossover exponent (φ) at the transition point is defined as, N s ∼ N φ , where N is the total number of monomers in the chain while N s is the number of monomers adsorbed on the surface.
In this paper, we have analytically solved fully directed self avoiding walk model to investigate dependency of adsorption transition point of an infinitely long linear semiflexible polymer chain when adsorption of the chain occurs on the constraint(s) and used the results so obtained to compare with the case when adsorption of a semiflexible polymer chain occurs on a flat surface [10,11].
The paper is organized as follows: In Sec. 2, lattice model of fully directed self avoiding walk is described for a linear semiflexible polymer chain confined in between the constraints for a particular value of spacing between the constraints under good solvent condition on a square lattice. In sub-section 2.1, we have discussed adsorption behaviour of the polymer chain when constraint A is having attractive interaction with the semiflexible polymer chain. Sub-section 2.2 is devoted to discuss the adsorption of the semiflexible polymer chain on the constraint B. While, in sub-section 2.3, expression of the partition function of the polymer chain is obtained for the case when the chain is having attractive interaction with the both the constraints. Finally, in Sec. 3 we summarize and discuss the results obtained.

Model and method
Lattice model of fully directed self-avoiding walk [9] on a square lattice has been used to analyze dependency of adsorption transition point of an infinitely long linear semiflexible homopolymer chain on its bending energy when the chain is confined in between two impenetrable stair shaped surface under good solvent condition (as shown schematically in figure 1). The directed walk model is restrictive in the sense that the angle of bending has unique value, that is 90 • for a square lattice and directedness of the walk amounts to certain degree of stiffness in the walks of the chain because all directions of the space are not treated equally. Since, directed self avoiding walk model can be solved analytically and therefore it gives exact value of the adsorption transition point of the polymer chain. We consider fully directed self avoiding walk (F DSAW ) model, therefore, walker is allowed to take steps along +x, and +y directions on a square lattice in between the constraints. The walks of the chain starts from a point O, located on an impenetrable surface and walker moves through out the space in between the two surface (as we have shown schematically in figure (1) that a walk of the polymer chain confined in between two surface for one value of separation n(= 3) between the two surface or the constraints).
The stiffness of the chain is accounted by associating a Boltzmann weight with bending energy for each turn in the walk of the polymer chain. The stiffness weight is k(= exp(−βǫ b ); where β = 1 k b T is inverse of the temperature, ǫ b (> 0) is the energy associated with each bend in the walk of the Figure 1: This figure shows a walk of an infinitely long linear semiflexible polymer chain confined in between two constraints (impenetrable stair shaped surface). All walks of the chain starts from a point O on the constraint. We have shown three different cases viz. (i), (ii) and (iii) having separation (n) between the constraints along an axis three monomers (steps). The separation between the constraints have been defined on the basis of the fact that how many maximum number of steps a walker can successively move along any of the +x or +y direction. In the case 1(i), the constraint A is having attractive interaction with the monomers of the chain, in 1(ii) only constraint B is having attractive interaction with the monomers of the chain while in 1(iii) both constraints are shown to have attractive interaction with the monomer of the polymer chain.
chain, k b is Boltzmann constant and T is temperature). For k = 1 or ǫ b = 0 the chain is said to be flexible and for 0 < k < 1 or 0 < ǫ b < ∞ the polymer chain is said to be semiflexible. However, when ǫ b → ∞ or k → 0, the chain has shape like a rigid rod. The partition function of a surface interacting semiflexible polymer chain can be written as, all walks of N steps where, N b is the total number of bends in a walk of N steps (monomers), N s is number of monomers in a N step walk, lying on the surface, g is the step fugacity of each monomer of the chain and ω is Boltzmann weight of monomer-surface attraction energy.

Adsorption of the semiflexible polymer chain on the constraint A
The partition function of an infinitely long linear semiflexible polymer chain confined in between the constraints (as shown schematically in figure 1-i) and having attractive interaction with the constraint A can be calculated using the method of generating function technique. The components (as shown in figure 2) of the partition function, Z A 3 (k, ω 1 ) (we have used here suffix three because in figure 1-i case maximum step that a walker can move successively in one particular direction is three and ω 1 is Boltzmann weight of attraction energy between monomers and the constraint A) of the chain can be written as, where, s 1 = ω 1 g. and On solving Eqs. (2-7), we get the expression for X A 1 (k, ω 1 ) and Y A 2 (k, ω 1 ). In obtaining the expression for X A 1 (k, ω 1 ) and Y A 2 (k, ω 1 ), we have solved a matrix of 2nX2n (n = 3, for present case i. e. figure 1-i). Thus, we have exact expression of the partition function for an infinitely long linear Figure 2: The components of the partition function is shown graphically in this figure. Term X A m (3 ≤ m ≤ n) indicates sum of Boltzmann weight of all the walks having first step along +x direction and suffix n indicates maximum number of steps that walker can successively take along +x direction. Similarly, we have defined Y A m , where first step of the walker is along +y direction. In this figure, (ii) and (iii) graphically represents recursion relation for Eqs.(2&6) respectively. semiflexible polymer chain confined between the constraints and having attractive interaction with the constraint A (as shown in figure 1-i) is written as, ) From singularity of the partition function, Z A 3 (k, ω 1 ), we obtain critical value of monomer-constraint A attraction energy, ω c1 = √ 1−2k 2 g 2 −k 2 g 4 +k 4 g 4 √ k 2 g 2 +k 2 g 4 −2k 4 g 4 +k 2 g 6 −2k 4 g 6 +k 6 g 6 , required for adsorption of an infinitely long linear semiflexible polymer chain on the constraint A. We have obtained the value of ω c1 = 1, when we substitute value of g c in the expression of ω c1 corresponding to all possible values of k[= exp(−βǫ b )] or bending energy ǫ b for which an infinitely long linear semiflexible polymer chain can be polymerized in between the constraints.

Adsorption of the semiflexible polymer chain on the constraint B
The partition function of an infinitely long linear semiflexible polymer chain confined in between the constraints (as shown schematically in figure 1ii) and having attractive interaction with the constraint B is calculated following the method discussed in above subsection. The components of the partition function, Z B 3 (k, ω 2 ) (where, ω 2 is Boltzmann weight of attractive interaction energy between the monomers of the chain and the constraint B) of the chain can be written as, where, and On solving Eqs. (9)(10)(11)(12)(13)(14), we get the expression for X B 1 (k, ω 2 ) and Y B 2 (k, ω 2 ). In obtaining the expression for X B 1 (k, ω 2 ) and Y B 2 (k, ω 2 ), we have to solve a matrix of 2nX2n (n = 3, for figure 1-ii case). Thus, we obtained exact expression of the partition function for an infinitely long linear semiflexible polymer chain confined between the constraints and having attractive interaction with the constraint B (as shown in figure 1-ii) is as, ) From singularity of the partition function, Z B 3 (k, ω 2 ), we could obtain critical value of monomer-constraint B attraction energy, ω c2 = √ 1−2k 2 g 2 −k 2 g 4 +k 4 g 4 √ k 2 g 2 +k 2 g 4 −2k 4 g 4 +k 2 g 6 −2k 4 g 6 +k 6 g 6 = ω c1 , required for adsorption of an infinitely long linear semiflexible polymer chain on the constraint B. In this case too, we have value of ω c2 = 1, for all possible values of bending energy or stiffness of the semiflexible polymer chain.

Adsorption of the semiflexible polymer chain on the constraints A&B
The partition function of an infinitely long linear semiflexible polymer chain confined in between the constraints (as shown schematically in figure 1iii) and having attractive interaction with both the constraints (A&B) is calculated following the method discussed above. The components of the partition function, Z C 3 (k, ω 3 , ω 4 ) of the chain can be written as, where, s 3 = ω 3 g. and On solving Eqs. (16-21), we get the expression for X C 1 (k, ω 3 , ω 3 ) and Y C 2 (k, ω 3 , ω 4 ). In obtaining the expression for X C 1 (k, ω 3 , ω 4 ) and Y c 2 (k, ω 3 , ω 4 ), we have solved a matrix of 2nX2n (n = 3, for figure 1-iii case). Thus, we have exact expression of the partition function for an infinitely long linear semiflexible polymer chain confined between the constraints and having attractive interaction with the constraints (as shown in figure 1-iii) is written as, From singularity of the partition function, Z C 3 (k, ω 3 , ω 4 ), we obtain critical value of monomer-constraints attraction energy, required for adsorption of an infinitely long linear semiflexible polymer chain on the constraints A when both the constraints have attractive interaction with the chain.
On substitution of following value of ω 4 in Eq. (23) to get value of ω c3 = 1.
√ k 2 g 2 +k 2 g 4 −2k 4 g 4 +k 2 g 6 −2k 4 g 6 +k 6 g 6 = ω c2 , The method discussed above can be used for different values of n and size of the matrix needed to solve in calculating partition function of the chain confined in between the constraints is 2nX2n. We have calculated exact expression of the partition function for n (3 ≤ n ≤ 19).
We have found that adsorption transition point of an infinitely long linear semiflexible polymer chain on the constraint A, B and simultaneously on both the constraints A&B has value unity. This fact is true for all the chosen values of k or bending energy (and 3 ≤ n ≤ ∞) for which an infinitely long polymer chain can be polymerized in between the constraints.

Result and discussion
We have considered an infinitely long linear semiflexible homopolymer chain confined in between two impenetrable stair shaped surface (constraint) on square lattice under good solvent condition. We have used fully directed self avoiding walk model to study adsorption phase transition behaviour of the polymer chain on any of the two constraints (A& B) and simultaneous adsorption of the polymer chain on both the constraints A&B. The generating function technique is used to solve the model analytically and exact expression of the partition function of surface interacting semiflexible polymer chain is obtained for different values of spacing (3 ≤ n ≤ 19) between the constraints.
It has been found that adsorption of the polymer chain occurs on the constraints at a value ω c1 = ω c2 = ωc3 = 1 for all possible values of k or bending energy of the chain for which an infinitely long linear semiflexible polymer chain can be polymerized in between the constraints. The critical value of ω is unity for all the considered cases by us for different values of spacing between the constraints (3 ≤ n ≤ 19). This result is obvious because walks of the chain is directed along the constraint(s), therefore, partition function of the chain is dominated by walks lying on the constraints and adsorption of the chain will occur even there is no attraction between constraint(s) and monomer of the chain.
However, in the case of adsorption of an infinitely long linear semiflexible polymer chain on a flat surface, the adsorption transition point is found to depend on the bending energy or stiffness of the chain. In this case, stiffer chains adsorption occurs at a smaller value of monomer surface attraction than the flexible polymer chain [10,11].