Effects of porous media on the phase behaviour of polypeptide solutions

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Condensed Matter Physics, 2022, Vol. 25, No. 3, 33602: 1–13

DOI: 10.5488/CMP.25.33602

http://www.icmp.lviv.ua/journal

Effects of porous media on the phase behaviour of

polypeptide solutions

V. I. Shmotolokha ∗

, M. F. Holovko

Institute for Condensed Matter Physics, 1 Svientsitskii St., 79011 Lviv, Ukraine

Received April 19, 2022, in ﬁnal form July 13, 2022

The generalized van der Waals equation for anisotropic ﬂuids in porous media, proposed by the authors in

previous works, is used to describe the effect of porous media on the phase behavior of polypeptide solutions.

By introducing the temperature dependence for the depth of the potential well and the geometric parameters of

the spherocylinder, the main features of phase behavior of the polypeptide poly (𝛾-benzyl-𝐿-glutamate) (PBLG)

in a solution of dimethylformamide, including the existence of two nematic phases, is reproduced. It is shown

that the presence of a porous medium shifts the phase diagram to the region of lower densities and lower

temperatures.

Key words: hard spherocylinder ﬂuid, porous material, scaled particle theory, isotropic-nematic transition,

polypeptides

1. Introduction

Physico-chemical research on the physics of polymers has long focused mainly on the study of linear

ﬂexible polymers, the ﬂexibility of which depends on the chemical structure of the polymer, its chemical

functional groups and environment [1, 2]. Stiﬀ and semi-stiﬀ polymer systems have also attracted

considerable attention in recent decades. Macromolecules of such polymers are quite stiﬀ or contain stiﬀ

functional groups, as a result of which the polymer systems may exhibit liquid crystal properties [3, 4].

As a result, polymer systems can be characterized by a wide range of phase transitions, which make their

properties diﬀerent from the properties of ﬂexible polymers.

Liquid crystal orientation was ﬁrst observed in biopolymers, in particular in rod-like viruses such

as tobacco mosaic viruses [5, 6]. There are also a number of other biological polymers that exhibit

liquid crystal properties. Among them are peptide compounds whose molecules consist of two or more

residues of 𝛼-amino acids which are connected in an unbranched chain by a covalent peptide bond -C(O)-

N(H)-. Depending on the number of amino acid residues in the chain, polymers belong to the family of

polypeptides or proteins. Most polypeptide and protein molecules are fairly stiﬀ rod-like molecules. In

particular, proteins such as collagen, spectrin, myosin, actin and keratin have a stiﬀ or semi-stiﬀ rod-like

structure, which is essential for their biological functions [7].

In aqueous solutions, polymers are studied as polyelectrolytes. Therefore, along with the shape of

molecules, electrostatic interactions also play an important role in them. The study of nonionic synthetic

helix polypeptides in nonpolar organic solvents was initiated by [8, 9], which showed an important role

of nonspherical shape of molecules in the formation of the liquid crystal phase. Particular attention was

paid to the study of the phase behavior of a solution of a polypeptide of poly (𝛾-benzyl-𝐿-glutamate)

(PBLG) in dimethylformamide (DMF), which is characterized by signiﬁcant regions of nematic order

stability and interesting phase coexistence of two liquid crystal states with noticeable concentrations of

polypeptides. Key experimental data of the research of the phase behavior of PDLG solutions in DMF

was obtained in the Miller group [10–14], which use various experimental techniques, such as proton

∗Corresponding author: shmotolokha@icmp.lviv.ua.

This work is licensed under a Creative Commons Attribution 4.0 International License. Further distribution

of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

33602-1

arXiv:2210.00998v1 [cond-mat.soft] 3 Oct 2022

V. I. Shmotolokha, M. F. Holovko

nuclear magnetic resonance (NMR), polarization spectroscopy, diﬀerential scanning calorimetry (DSC)

and viscometer. The phase behavior of the PBLG solution in DMF was also discussed in detail in [15, 16].

For the theoretical interpretation of the obtained phase behavior of the PBLG solution in DMF in

the ﬁrst works [10, 11], the Flori lattice model was used [17, 18]. However, despite some qualitative

agreement between theoretical and experimental data, at a discrete representation of the solution in a

lattice consideration and unnatural interpretation of rod-like molecules, it is diﬃcult to make a direct

comparison between the lattice model and real macromolecules. A more realistic non-lattice model for

describing the phase behavior of a PBLG solution in DMF was ﬁrst proposed relatively recently in

the Jackson group [19]. In the proposed model, the PBLG solution is considered as a system of hard

spherocylinders with attraction in the form of an anisotropic potential well. This model is described in

the framework of the previously developed [20] approach, which combines the Onsager [21] approach

for describing rather long hard spherocylinders with the van der Waals approach that account for the

attractive interactions [22]. To consider the change in the conformation of the macromolecule PBLG, the

authors of [19] introduced the temperature dependence of the parameters of the spherocylinder and the

depth of the potential well with the appropriate ﬁtting parameters. As a result, the authors were able to

reproduce the experimental phase diagram quite well. However, to describe the orientational ordering, the

authors used a trial function in the Onsager form [21], which signiﬁcantly overestimates the orientational

ordering [23].

It should be indicated that all description of isotropic-nematic transition in non-spherical particles

since Onsager’s theory [21] can be considered as diﬀerent variants of perturbative density functional

theory type. In all cases, the free energy of the considered systems is formulated with some approximation

as a functional of the orientation-dependent singlet distribution function 𝑓(Ω). The minimization of the

free energy (or grand canonical potential) of the singlet distribution function leads to a non-linear integral

equation for singlet distribution function, which can be solved numerically or by using the trial function

with the coeﬃcients, which can be found by minimizing the free energy. An example of such an approach

was recently presented in [24], where for a description of thermodynamics of anisotropic systems of

particles with steric interactions there was used a formalism of functional density, which was founded

by [25, 26] and then generalized in the theories by van der Waals.

A well-studied phase diagram of a solution of PBLG in DMF could work as a good base model

for understanding the phase behavior in biopolymers. The next step should be to take into account

the biological environment, which is often seen as a porous environment [27]. This is the purpose

of the present work. The theoretical basis of the work is based on the works [28–30], in which the

generalization of van der Waals equation on anisotropic ﬂuids in porous media is given. In contrast

to [20], the orientational ordering in the system is described through a corresponding integral equation

for the singlet distribution function, which is found by minimizing the free energy functional of the

system and its numerical solution on the basis of generalization [31].

2. Theory

As in [19, 20], we consider a model of hard spherocylinders with length 𝐿1and diameter 𝐷1with

attraction, which we choose to describe as an anisotropic potential well

𝑢attr (𝑟, Ω1,Ω2)=−[𝜖0+𝜖2𝑃2(cos 𝛾12)], 𝛾1𝐷1> 𝑟 > 𝜎 (Ω1,Ω2,Ω𝑟),

0, 𝑟 < 𝜎(Ω1,Ω2,Ω𝑟), 𝑟 > 𝛾1𝐷1,(2.1)

where 𝜖0and 𝜖2characterize isotropic and anisotropic parts of the attractive interaction respectively,

𝛾1=1+𝐿1/𝐷1,𝑃2(cos 𝛾12)=1/2(3 cos2𝛾12 −1)is the second order Legendre polynomial of relative

orientation, 𝛾12 is the angle between two principal axes of two spherocylinders. We hope that the reader

will not confuse the angle 𝛾12 with notation 𝛾1=1+𝐿1/𝐷1.𝜎(Ω1,Ω2,Ω𝑟)is the orientation-dependent

contact distance of two particles, Ω1and Ω2are orientations of two particles 1 and 2, Ω = (𝜃, 𝜙)

denotes the orientations of particles, which are deﬁned by the polar angles 𝜃and 𝜙.Ω𝑟is deﬁned by the

angles between the ﬁxed system of coordinates and the system of coordinates, which moved with two

considered particles. In terms of 𝜎(Ω1,Ω2,Ω𝑟), the repulsive part of the interaction 𝑢rep (𝑟12,Ω1,Ω2)for

hard particles can be represented in the form

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Effects of porous media on the phase behaviour of polypeptide solutions

𝑢rep(𝑟12,Ω1,Ω2)=∞,for 𝑟12 < 𝜎(Ω1,Ω2,Ω𝑟)

0,for 𝑟12 > 𝜎(Ω1,Ω2,Ω𝑟).(2.2)

Like the potential of interaction, thermodynamic functions of the considered model can also be repre-

sented as the sum of two terms, the ﬁrst from the hard spherocylinders and the second from the attractive

interaction. In particular, the free energy of the system

𝐹

𝑁1𝑘𝑇 =𝐹0

𝑁1𝑘𝑇 +𝐹attr

𝑁1𝑘𝑇 ,(2.3)

where 𝑁1is the total number of ﬂuid particles, 𝑘is the Boltzmann constant, 𝑇is the temperature, 𝐹0

is the contribution from hard spherocylinders, 𝐹attr is the contribution from the attractive part of the

interaction.

To describe the contribution from hard spherocylinders, we use the scale particle method, which was

recently developed in our group to describe a ﬂuid of hard spheres in a porous medium [32–35] and

was generalized to the case of a ﬂuid of hard spherocylinders in a porous medium [28, 36, 37]. The key

point in the scale particle method for a spherocylinder system is the introduction of a scale cylinder with

replaceable dimensions

𝐷𝑠=𝜆𝑠𝐷1, 𝐿𝑠=𝛼𝑠𝐿1.(2.4)

The method is based on the exact calculation of the chemical potential of this large-scale particle at

𝜆𝑠→0and 𝛼𝑠→0and its combination with thermodynamic consideration of the large-scale particle

of macroscopic size. However, the standard scaled particle theory (SPT) of the ﬂuids in the presence of

porous media contains a subtle discrepancy, which appeared in the case when the size of matrix particles

is essentially larger than the size of ﬂuid particles. This discrepancy was eliminated in [33, 34] as part

of a new approach named SPT2. The expressions, which are derived from this approach, include two

parameters that deﬁne the porosity of a matrix. The ﬁrst parameter is named geometrical porosity 𝜙0. It

characterizes the free volume, which is not occupied by matrix particles. The second parameter is named

a porosity of the probe particle 𝜙. It is deﬁned by the chemical potential of a ﬂuid in the limit of inﬁnite

dilution and has a meaning of probability to ﬁnd a particle of ﬂuid in an empty matrix. It was shown that

the SPT2 agrees well with the data of computer simulation at low densities of ﬂuid, while at medium and

at high densities we could obtain an essential diﬀerence. This diﬀerence becomes especially important

when the packing fraction of a ﬂuid 𝜂1reaches the values close to the porosity of the probe particle

𝜙, because the obtained expressions diverge at 𝜂1→𝜙. As a result, various approximation schemes

were propose. The accuracy of them was verifed by comparing theoretical results with the corresponding

computer simulation data. Herein below, we concentrate on the SPT2b1 approximation the results of

which reproduce computer simulation data quite well. Omitting the results of the calculations, the details

of which are given in our previous publications [28, 36, 37], we present the ﬁnal results for the chemical

potential, pressure and free energy, respectively.

𝛽𝜇ex

1−𝜇0

1SPT2b1

=𝜎(𝑓) − ln(1−𝜂1/𝜙0) + [1+𝐴(𝜏(𝑓))] 𝜂1/𝜙0

1−𝜂1/𝜙0

+𝜂1(𝜙0−𝜙)

𝜙0𝜙(1−𝜂1/𝜙0)+1

2[𝐴(𝜏(𝑓)) + 2𝐵(𝜏(𝑓))] (𝜂1/𝜙0)2

(1−𝜂1/𝜙0)2

3𝐵(𝜏(𝑓)) (𝜂1/𝜙0)3

(1−𝜂1/𝜙0)3,(2.5)

𝛽𝑃

𝜌1SPT2b1

1−𝜂1/𝜙0

𝜙0

𝜙+𝜙0

𝜙−1𝜙0

𝜂1ln 1−𝜂1

𝜙0

+𝐴(𝜏(𝑓))

𝜂1/𝜙0

(1−𝜂1/𝜙0)2+2𝐵(𝜏(𝑓))

(𝜂1/𝜙0)2

(1−𝜂1/𝜙0)3,(2.6)

33602-3

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CondensedMatterPhysics,2022,Vol.25,No.3,33602:113DOI:10.5488/CMP.25.33602http://www.icmp.lviv.ua/journalEffectsofporousmediaonthephasebehaviourofpolypeptidesolutionsV.I.Shmotolokha*,M.F.HolovkoInstituteforCondensedMatterPhysics,1SvientsitskiiSt.,79011Lviv,UkraineReceivedApril19,2022,inﬁnalformJuly1...

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