Swelling Pressure Induced Phase-Volume Transition in Hybrid

Biomacromolecules 2003, 4, 1818-1826

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Swelling Pressure Induced Phase-Volume Transition in Hybrid Biopolymer Gels Caused by Unfolding of Folded Crosslinks: A Model Karel Dusˇ ek,*,†,§ Miroslava Dusˇ kova´ -Smrcˇ kova´ ,†,‡ Michal Ilavsky´ ,†,‡ Russell Stewart,§ and Jindrˇich Kopecˇ ek|,§ Institute of Macromolecular Chemistry, Academy of Sciences of the Czech Republic, 162 06 Prague 6, Czech Republic, Department of Macromolecular Physics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic, Department of Bioengineering, University of Utah, Salt Lake City, Utah 84112, and Department of Pharmaceutics and Pharmaceutical Chemistry, University of Utah, Salt Lake City, Utah 84112 Received July 2, 2003; Revised Manuscript Received September 2, 2003

A thermodynamic model is proposed describing swelling changes and swelling transitions of hybrid gels in which domains of folded chains are chemically built in as cross-links. These folded domains can be unfolded to random coils by osmotic forces produced by the synthetic gel matrix. Uncoiling takes place if the osmotic force acting on the cross-links exceeds the critical value. By unfolding, a new interacting surface is exposed to interactions and affects the swelling pressure. The chains of the folded domains may have ionized groups. The model is based on mean-field statistical-thermodynamic treatment of swelling of polyelectrolyte gels with finite extensibility of network chains. This study is related to hybrid hydrogels with built in protein motifs. A continuous change in external variables increasing the degree of swelling of the hydrogel brings about an abrupt increase in volume (transition) of the gel. The position and magnitude of the transition depend on structural parameters of the hybrid gel, such as fraction of the folded domains in the gel, degree of ionization of chains in the domain, presence of additional chemical cross-links, or degree of dilution at gel formation. Two options for reversibility of the changes are considered: (a) unfolding is irreversible and deswelling proceeds along other curve than swelling and (b) swelling is reversible when the osmotic force decrease below the critical value. In the latter case, swelling changes are described by a closed loop with two transitions. Under certain conditions (high dilution at network formation and sufficiently high degree of ionization of chains of the folded domains), a transition appears known as the collapse transition induced by balance of hydrophobic and hydrophillic interactions. This collapse transition induces the folding transition by which the folded domains are reformed. Introduction The equilibrium swelling degree of polymer gels in water or in other solvents is determined by the existence of a covalent or physical polymer network or combination of both. The polymer network can be composed of synthetic polymer chains, usually disordered, or of biopolymer (protein) motifs. The covalently cross-linked biopolymer motifs such as elastin (cf, e.g., refs 1 and 2) are used as extracellular matrixes of controlled mechanical properties or cross-linked actin and myosin like gel machines.3 The majority of physical gels are of a natural origin; they are obtained by association of biopolymer motifs and can dissociate or reform upon change of external conditions.7 Gelatin gels can serve as an example. Also some synthetic gels are of a physical nature: crosslinks are generated reversibly, for example, by crystallization, complex formation, or numerous entanglements. * To whom correspondence should be addressed. † Academy of Sciences of the Czech Republic. ‡ Charles University. § Department of Bioengineering, University of Utah. | Department of Pharmaceutics and Pharmaceutical Chemistry, University of Utah.

In the newly developed class of polymer gels called hybrid biopolymer gels, ordered biopolymer motifs are embedded in synthetic disordered hydrogels and chemically bonded to the (hydro)gel chains. The network can be formed by association of the motifs or by chemical cross-linking.4,5 Hybrid biopolymer gels bridge synthetic disordered gels and biopolymer gels. Modulation of synthetic gels by protein motifs has been used since some time ago, for instance, for building-in into the gel enzymatically cleavable units.6 The hybrid biopolymer gels are distinguished by several special features such as (a) the role of the disordered synthetic network in transmitting the effect of structural changes of the biopolymer motifs or transmitting the effect of external variables acting on motifs or (b) by the existence of additional interactions between the motifs and segments of the hydrogel polymer. Thus, the disordered network can either play a passive role by transmitting structural changes generated by the biopolymer motifs and resulting in a macroscopic response of the gel, or the osmotic force generated by the synthetic network acts on the motifs and can induce their

10.1021/bm034219s CCC: $25.00 © 2003 American Chemical Society Published on Web 10/17/2003

Phase-Volume Transition in Hybrid Biopolymer Gels

conformational transition. In this contribution, we will deal with the latter case. Several factors determine the swelling degree of a disordered covalently cross-linked polymer network, such as polymer-solvent interactions, cross-link density, conditions at network formation, and, in the case of polyelectrolytic systems, degree of ionization, formation of ion pairs, or presence of external electrolyte. Externally, the degree of swelling is controlled by temperature, solvents nature and composition, pH, addition of nonionizable and ionizable compounds, etc. A certain balance of attractive and repulsive interactions in disordered gels may lead to a special kind of compensation, such that two network phases of different degrees of swelling can coexist. The transition between the two states of the network characterized by different degree of swelling, already predicted in 19688 and discovered by Tanaka 10 years later,9 is called the Volume phase transition or collapse transition (cf., e.g., refs 10-12). Gels exhibiting this transition are called responsive gels; they can undergo swelling transitions induced by changes in temperature, pH, electric field, by irradiation, addition of specific chemicals, etc. Oscillating changes have also been described.13 For these types of transitions, the presence of biopolymer motifs and their structural rearrangement is not necessary. The hybrid biopolymer gels, such as those described in ref 4, contain folded elements (motifs) that unfold when a certain limiting stress is applied to the chain ends. A muscle protein titin is a single chain giant molecule composed of 27 000 amino acid units. It contains more than 200 immunoglobulin (Ig) and fibronectin domains. The folded element, a repeating Ig domain, is a sandwich of two antiparallel β sheets held together by hyprophobic interactions between the β sheets in the core of the domain and by hydrogen bonds between the β strands. When individual titin molecule is stretched, the Ig domains unfold one after another. Using atomic force microscope or optical tweezers, a periodic (sawlike) force-extension dependence was observed14 corresponding to successive unfolding of Ig domains. Thermodynamic and dynamics simulation models of the unfolding behavior of titin were described.14-17 A similar unfolding behavior was observed for ribozyme;18 the contraction following the release of the stress is monotonic. In the hybrid gel described in ref 4, the Ig domains were attached to the chains of a polyacrylamide gel through a strong complex between histidine tagged Ig units and specific sites on polyacrylamide chains. The external force applied to the folded chain ends was caused by the osmotic pressure produced by swelling of a hydrophilic polyacrylamide network. It was expected that unfolding of Ig domains as a result of an increase in osmotic pressure would lead to an increase in the degree of swelling. The experiments have shown that a change of temperature produced a large and irreversible change in the degree of swelling connected with Ig denaturation. However, this transition has not been studied quantitatively. The model described below based on statistical thermodynamics of polymer solvent interactions and polymer chain elasticity is proposed for swelling changes of a gel in which

Biomacromolecules, Vol. 4, No. 6, 2003 1819

Figure 1. Structure of a hybrid gel with cross-links in the folded and unfolded states.

a part of the cross-links contains folded domains. These folded domains unfold after a limiting osmotic force acting on cross-link domain ends is reached. The predicted swelling changes can be continuous or discontinuous depending on the nature of the interacting sites and the density of the crosslinks. A simplified version of this model was already proposed in ref 19. In this new version, several changes have been made, so that the model is more realistic. The limited extensibility of network chains was considered and the dependence of the force acting on cross-link was also somewhat modified. Consequences of possible reversibility or irreversibility of the unfolding process on swelling equilibria were also examined. It was discovered that the swelling transition due to unfolding can be sometimes accompanied by classical collapse transition of network chains. Swelling of Gels with Folded and Unfolded Domains: Theory Qualitative Description of the Model. The structure of a hybrid biopolymer gel and the transition of the domains from the folded to unfolded state are schematically shown in Figure 1. By solvation (hydration) of the hydrogel chains HP, a swelling pressure is generated. FX are folded domains with a small interacting surface acting as tetrafunctional crosslinks. In addition to FX, also conventional chemical crosslinks, CX, may exist in the network. Before unfolding, at equilibrium the swelling pressure is counterbalanced by the elastic force arising from stretching chains HP. When the osmotic pressure increases due to changes of external conditions, the tensile force acting at the ends of the crosslinks also increases. When this force exceeds a limiting value, a fraction of the cross-links unfolds and the new, inner surface of the folds is exposed to interaction with the solvent molecules or hydrophilic polymer segments. If the osmotic pressure and the force acting on the cross-link further increases, all folded domains are unfolding one after another, until all of them unfold. This is manifested by a jump in swelling degree. Moreover, by unfolding one tetrafunctional cross-link FX is transformed into two trifunctional crosslinks UX. The UX junctions are connected with a UP chain. It has been shown20 that disordered (denaturated) biopolymer motifs behave as randomly coiled chains and that their conformation can be, with reasonable approximation, described by a so-called equivalent chain (charged). The length of the statistical segment of the UP chain is generally different from that of the hydrogel polymer HP. Therefore,

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Dusˇ ek et al.

the problem of swelling of networks containing both synthetic chains and biopolymer motifs can be treated within the model of polyelectrolyte networks composed of statistically equivalent chains of finite extensibility. Gibbs Energy Changes. The Gibbs energy change on swelling of a gel with ionizable groups, ∆Gsw, is considered to be composed of the following contributions:21,22 mixing of polymer segments with solvent molecules, ∆Gmix, deformation of the network, ∆Gdef, electrostatic repulsion due to fixed charges, ∆Gelst, and effect of small ions, ∆Gion, (Donnan effect) ∆Gsw ) ∆Gmix + ∆Gdef + ∆Gelst + ∆Gion

( )( ) 1 ∂∆Gsw V h 1 ∂N1

T,p,N2

1 RT ) - ∆µ1 ) - ln a1 ) 0 (2) V h1 V h1

where V h 1, N1, and a1 are respectively the molar volume, number of moles, and activity of the solvent. Differentiation of contributions to ∆Gsw with respect to N1 gives the respective contributions to Πsw Πsw ) Πmix + Πdef + Πion + Πelst

(3)

The mixing contribution is based on the Flory-Huggins and Tompa23 theory of polymer solutions formulated for surface fractions of the polymer ψ2 Πmix ) -(RT/V1)[ln(1 - ψ2) + ψ2 + χ(ψ2)ψ22]

(4)

Usually, the surface fraction ψ2 is considered to be equal to the volume fraction φ2. This corresponds roughly to the situation when all chains are unfolded. However, both fractions may strongly differ when the biopolymer domains are folded. The Flory-Huggins interaction parameter χ can be concentration dependent. One can also work with the interaction function g(φ2) or g(ψ2) (cf., ref 24 for interrelations between χ and g). The deformation contribution is based on the Flory-Erman theory of rubber elasticity (junction-fluctuation model),26 extended to cover finite extensibility of network chains. The effect of finite extensibility becomes important at high swelling degrees. To keep the separability of contributions along the x, y, and z axes, the perturbation method of power series expansion of the Langevin function was used.21 Because the deformation is a function of distances between elastically active cross-links, Πdef is a function of φ2 and not ψ2

{

n)

(1)

The equilibrium swelling degree expressed by the volume fraction of the polymer in the swollen network, φ2, is obtained by the solution of the equation for the swelling pressure, Πsw, or the chemical potential of the solvent, ∆µ1 Πsw ) -

concentration of EANCs, νed, φ0 is the volume fraction of polymeric components at network formation (1 - φ0 is the volume fraction of diluent and sol), Af is the front factor varying from (fe - 2)/fe (phantom network limit) to 1 (affine network limit), and the corresponding values of the factor B are 0 and 2/fe, where fe is the average number of strands with infinite continuation per elastically active cross-link. The number of statistical segments of a hydrogel EANC, n, is related to the concentration of EANCs and the molecular weight of the statistical segment, Mst, as

3 Πdef ) -RTνed (Afφ02/3 φ21/3 - Bφ2) + φ04/3 φ2-1/3n-1 + 5 99 2 -1 -2 513 8/3 -5/3 -3 φ φ n + φ0 φ2 n + ... (5) 175 0 2 875

}

where νed is the concentration of elastically active network chains (EANC) in unit dry volume, n is the number of statistical segments per EANC and thus a function of the

Fhg νedMst

The value of Mst can be changed after unfolding. The contribution by electric charges (fixed ions) on the network chain extension is based on the Katchalsky-Lifson model and offers the following relation for the contribution Πsw: νedNAZi2e2φ24/3 2.5A - ln(1 + A) Πelst ) RT 3D(rj02 φ0-2/3)1/2 1 + A

(

where A ) 6r/κrj02 )

(

φ02/3 φ2-2/3DkTM0

)

(6)

)

1/2

πNAe2jr02(2M0c- + iFφ2)

For the contribution by counterions and coions, Πion, (Donnan effect) the corresponding equation reads

(

[(

iFφ2 Πion ) (RT/M0) iFφ2 - 2f-c-M0 1 + M0f-c-

) ]) 1/2

-1

(7)

In eqs 6 and 7, M0 is the molecular weight of the monomer unit, i is the degree of ionization of the polymer (fraction of charged groups), F is the density of the dry polymer, c- is the concentration of coions (small anions, if the polymer is negatively charged), f- is the activity coefficient of coions, r is the end-to-end distance of a network chain in the swollen state, jr02 is the mean-squared end-to-end distance of a network chain in the unperturbed state, D is the dielectric constant of the medium, NA is Avogadro’s number, Z is the degree of polymerization, e is the unit charge, and κ is the inverse of the Debye radius of the ion atmosphere. Relation between the Force Acting on the Cross-Link and the Osmotic Pressure. The force acting on a crosslink is the key quantity that determines if and when the crosslink unfolds. This force is determined by the deformation contribution to the swelling pressure Πdef. In the whole network, the retractive force due to the stretching of the chains is counterbalanced by other contributions, particularly Πmix and Πion, but inside the network, the cross-links are under stress produced by the stretching of the elastically active network chains (EANC). The force per EANC, fEANC is obtained from Πdef, (cf., eq 5) Ne Πdef ) -RTνedFdef(φ2) ) -RT Fdef(φ2) Vd

(8)



where Ne is the number of moles of EANCs and Vd is the volume of the dry gel; Fdef(φ2) is an abbreviation for the deformation function in eq 5. By passing to molecular parameters (subscript molec), one gets for fEANC -ΠdefAmolec,sw ) kBTVmolec,d-1 Vmolec,sw2/3Fdef(φ2) ) NeNA

fEANC )

kBTVmolec,d-1/3 φ2-2/3Fdef(φ2) (9) where Amolec,sw is the surface corresponding to one EANC in swollen state, NA is Avogardo’s number, and kB the Boltzmann constant. The force acting on cross-link, f *, is higher by a factor of fe/2 because there is one EANC per two bonds issuing from the elastically active cross-links. In this case, for folded domains and a close to perfect network, fe ) 4, and f* ) 2fEANC

(10)

The molecular volume of an EANC in dry state, Vmolec,dry Vmolec,dry )

1

(11)

νedNA

By substituting from eq 5, one gets f* )

2kBT 1/3

{A φ

2/3

f 0

3 φ2-1/3 - Bφ21/3 + φ04/3 φ2-1n-1 + 5

Vmolec,dry 99 2 -5/3 -2 513 8/3 -7/3 -3 φ φ φ φ2 n + ... (12) n + 175 0 2 875 0

}

are determined by eqs 2-7. It is not exluded but not likely that the unfolding process stops at xf > 0. Thus, xf is a variable determined by the algorithm of the decision making process of eq 14. Changes in the Swelling Degree as a Result of Unfolding. Unfolding of the cross-links causes changes in the magnitude and character of the interacting surface, concentration of EANCs, and, usually, the fraction of the ionizable groups. Therefore, several parameters of the swelling equation become functions of the fraction xf. As was stressed before, in a network with cross-links in the folded state, a part of the segments are excluded from contact with the solvent. It is only the surface of the folded domain which interacts. The surface fraction ψ2 defined is by xfb(wf/Ff)+(1 - xf)wf/Ff+whg/Fhg ψ2 ) (15) xfb(wf/Ff) + (1 - xf)wf/Ff + whg/Fhg + ws/Fs where wf, whg, and ws are weight fractions of the folded domains, the hydrogel polymer, and the solvent, respectively. Ff, Fhg, and Fs are the respective densities. The factor b determines the difference between the surface and volume fractions of the polymer in the swollen gel. It is equal to the fraction the interacting surface in the folded state relative to that after unfolding. The value of b must be lower than unity; for the current system, it is estimated to be of the order of 10-1. The volume fraction of the polymer φ2 is determined similarly as ψ2 in eq 15, but with b ) 1

For estimation of the net force acting on one cross-link, a comparison should be made relative to the reference state. In the reference state, there are on average no forces acting on cross-links and the network chains are in their states of ease.25 Such situation is approximately met at network formation. Therefore, the relative force f /rel

The mean interaction parameter of polymer segments with the solvent, χj, is contributed by the surface fractions of the hydrogel polymer, ψhg, folded protein domain, ψf, and unfolded protein domain, ψu

f /rel ) f * - f *(φ2 ) φ0)

χj ) ψhgχhg + ψfχf + ψuχu

(13)

defines the condition for unfolding of the folded domain. It starts if / f /rel gθ f rel,crit

φ2 ) ψ2(b ) 1)

where χhg, χf, and χu, are the respective contributions to the interaction parameter. The surface fractions are given by the relations

(14)

/ can be estimated from the AFM The value of f rel,crit experiment discussed in the Introduction section. However, this method is not very accurate for the determination of the absolute values of the force, and it is determined at a / can be relatively high deformation rate. Therefore, f rel,crit lower by more than 1 order of magnitude. The variable of structural transformations during the unfolding-folding process is the fraction of folded cross-links, xf. When the condition of eq 14 is met, the first fraction of folded domains (cross-links) ∆xffδxf unfolds and the fraction of folded domains changes from xf ) 1 to 1 - δxf. The condition of eq 14 is examined again. If this condition keeps holding (i.e., the acting force is equal or exceeds the critical value) with the new value of xf, and associated parameters, eventually all cross-links unfold and xf drops to zero (allor-none mechanism). The changes of associated parameters

(16)

ψhg )

whg/Fhg b(wf/Ff)xf (wf/Ff)(1 - xf) , ψf ) , ψu ) Sψ1 Sψ1 Sψ1 (17)

where Sψ1 ) whg/Fhg + b(wf/Ff)xf + (wf/Ff)(1 - xf) The differences between surface and volume fractions depend on the weight fraction of the initially folded domains in the system wf and on the contraction factor b, as well as on the fraction of domains that remain folded, xf. In the elastic deformation contribution, the concentration of EANCs, νed, changes because upon unfolding one tetrafunctional folded cross-link is transformed into two trifunctional ones. The total concentration of EANCs, νjed, is a sum of the contributions by conventional chemical crosslinks possibly existing in the hydrogel, (∆νd)hg, and cross-

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links contributed by protein domains, (∆νed)f νed ) (∆νed)hg + (∆νed)f

(18)

The contributions to the number of EANCs by possibly present chemical cross-links, (∆νed)hg, and by folded crosslinks, (∆νed)f, are determined by their respective volume fractions, Φhgd and Φfd, and concentrations of EANCs in the dry polymer. Thus, Φhgd and Φfd are equal to Φfd ) 1 - Φhgd )

wf/Ff whg/Fhg+wf/Ff

(19)

and (∆νed)hg ) Φhgd(νed)hg

(20)

The folded domain contribution is equal to Ff (∆νed)f ) Φhgd [xf + (3/2)(1 - xf)] M0f

(21)

The functionality of an elastically active cross-link, which determines the front factors A and B in Πdef, depends on the degree of unfolding because a tetrafunctional folded crosslink gives two trifunctional ones (cf., eq 21). The number of f-functional cross-links is related to the respective number of EANCs as ∆νe/(f /2). Therefore

fe )

∑i (fi∆νed)i/(fi/2) ∑i

(22) (∆νed)i/(fi/2)

which gives in the present case (tetrafunctional chemical cross-links) fe )

2(∆νed)hg + 2(∆νed)f(2 - xf) (∆νed)hg/2 + (∆νed)f[xf/2 + (4/3)(1 - xf)]

(23)

for (∆νed)hg ) 0, fe is only a function of xf 12(2 - xf)

fe )

take into account possibly existing ionizable groups in the hydrogel and on the folded domain surface. Also the electrostatic repulsion contribution described by Πelst depends on the fraction of the unfolded structure 1 xf through a change of dielectric constant, D, degree of ionization, i, or cross-link density, νed. Reversibility of Swelling Changes. There have been experimental indications that the increase of swelling of the gel with built-in Ig domains was not reversible,4 but the range of gel compositions and generated swelling pressures were rather limited. Thus, the reversibility of folding-unfolding transitions is not excluded even in this system. It is evident that swelling and deswelling curves cannot be identical because of the unfolding criterion of eq 14. In a certain range of φ2, the states of the system are not identical because the folded domain interior is excluded from or exposed to interactions. Starting at xf ) 1 and the increasing swelling pressure, by which swelling degree increases, the condition / is reached while xf is still equal to 1. Going the f /rel g f rel,crit opposite way, from higher to lower degrees of swelling, / / while f /rel(xf ) 1) < f rel,crit . Also, it f /rel(xf ) 0) > f rel,crit should be investigated whether the unfolding is absolutely irreversible due to strong steric hindrances or whether the transition from the unfolded to folded states can take place when the criterion of eq 14 is approached from the other direction (xf changes from 0 to 1). In the latter case, the swelling dependence is expected to have a closed loop form with two transitions, expansion and collapse, occurring under different external conditions (temperature, pH, ionic strength, etc.).

(8 - 5xf)

(24)

The relations given above are valid for perfect networks without sol and dangling chains or other topological defects. In the nonideal case, the quantities (∆νed)f and (∆νed)hg can be derived by application of the branching theory. Examples of handling of topological defects for specific hydrogels are given in ref 27. The protein domain upon unfolding (fraction 1 - xf) can contribute by a certain number of ionizable groups, Nion, per domain. The contribution by Φfd is proportional to the fraction of unfolded protein domains Φfd(1 - xf) and number of ionized groups ∆Πion ) Φfd(RT/M0f)[iNionFfφ2(1 - xf)]

(25)

In this equation, M0f is the molecular weight of the folded (protein) domain and i is the degree of ionization of the ionizable units. It is not difficult to modify the equation to

Simulation of Swelling Dependencies for a Simplified System To demonstrate the expected characteristic features of swelling behavior, the system was simplified in that 1. The contribution by Πelst was not considered because it was found to be small relative to other contributions 2. The coions were absent (c- ) 0, no added salt) 3. The interaction parameter χj was considered to be independent of concentration, and the contributions in eq 16 were considered to be equal: χhg ) χf ) χu; therefore, χj could be used as external variable expressing, for example, the effect of temperature 4. Neither the hydrogel chains nor the surface of the folded domains contained charged groups; all charged groups were in the interior of the protein domains and started contributing to interactions when the domain was unfolded The values of Πelst make usually a few percent of Πion and Tanaka28 or Onuki29 do not consider this effect at all. Simplification (2) can easily be relaxed because the dependence on the concentration of co-ions is simple. Condition (3) is apparently an oversimplification. For polar or aqueous systems, χj will certainly depend on polymer concentration and this dependence for ∂χj/∂φ2 > 0 will facilitate the transition. It is known that a strong concentration dependence of χ can be a sufficient condition for the appearance of a collapsed transition in nonionized gels (cf., e.g., ref 30). However, the purpose of this study was not to


Phase-Volume Transition in Hybrid Biopolymer Gels Table 1. Values of Parameters Usually Kept Constant in This Model Studies quantity

symbol

critical relative force necessary to unfold a cross-link, eq 14 molecular weight of folded protein domain, eq 25 fraction of surface of the cross-link domain exposed to interaction in the folded state, eq 15 molecular weight of the statistical segment of the hydrophilic synthetic polymer front factor Af in Flory-Erman rubber elasticity theory, eq 5 front factor B in Flory-Erman rubber elasticity theory, eq 5 specific gravities of the domain polymer and hydrogel polymer, respectively specific gravity of the solvent molar volume of the solvent

/ f rel, crit

M0f b Mst Af B Ff,Fhg Fs V1

value 10-14

2× 10 000 0.1 240 1 1 1.2 1.0 50

dimension N g/mol g/mol

g/cm3 g/cm3 cm3/mol

simulate the behavior of a specific system, but rather to find effects of changing structural parameters of the gel on the existence, position, and magnitude of this type of transition. In this model study, χj was used as an independent external parameter that governed changes in the degree of swelling, and the degree of swelling was characterized by the volume fraction of the polymer φ2 ) 1 - φ1. Thus, the swelling equilibrium is determined by the following equation:

{

-[ln(1 - ψ2) + ψ2 + χjψ22] - νed (Afφ02/3 φ21/3 - Bφ2) +

}

3 4/3 -1/3 -1 99 2 -1 -2 513 8/3 -5/3 -3 φ φ2 n + φ φ n + φ0 φ2 n + 5 0 175 0 2 875 Φfd(RT/M0f)[iNionFfφ2(1 - xf)] ) 0 (26) For higher values of χj, the force acting on the cross-links is / ) and xf remains lower than the critical force (f /rel < f rel,crit equal to 1. The swelling degree increases as a result of a / decrease in χj. When f /rel reaches f rel,crit , the effect of / variation xf on f rel is examined while keeping the values of all other parameters independent of xf constant. If (f /rel / ) > 0, unfolding of the cross-links starts. For model f rel,crit systems described here, unfolding promotes further increase in the degree of swelling and force per cross-link f /rel. Therefore, all folded domains eventually unfold by the allor-none mechanism by keeping χj constant until the state of xf ) 0. The swelling degree changes abruptly, i.e., the / )g transition is discontinuous. If the condition (f /rel - f rel,crit 0 is never met at all values of χj, the swelling degree is controlled by xf ) 1 and the swelling curve is continuous. / ) < 0 for all Also, no transition is expected if (f /rel - f rel,crit values of χj investigated. A few examples have been selected to illustrate the effect of gel structure variables on swelling behavior: 1. effect of the weight or volume fractions of hydrogel and folded domains, characterized by the weight ratio a ) whg/wf which also determines the volume fractions Φfd and Φfd. In the absence of conventional chemical cross-links in the hydrogel, the value of a also determines the concentration of EANCs 2. effect of dilution at network formation given by the value of φ0 3. effect of concentration of ionized groups in the built-in protein component determined by the product iNion 4. effect of the presence of chemical cross-links in the hydrogel component 5. effect of values of front factors Af and B in Πdef, i.e., difference between phantom and affine network behavior

Figure 2. Swelling transition by unfolding of cross-links: effect of weight fraction of folded domains; a ) whg/wf weight ratio of hydrogel to folded cross-links indicated; volume fraction of polymer at network formation, φ0 ) 0.3; number of ionizable groups per unfolded crosslink, Nion ) 20; contribution to concentration of elastically active network chains by cross-links in the synthetic part of hydrogel, (νe)hg ) 0; other parameters as in Table 1.

6. effect of directions of changes of χj 7. effect of impossibility of back-folding of protein domains The values of parameters usually kept constant in this model studies are listed in Table 1. [The molar volume of the solvent was chosen to be equal to 50 cm3/mol, instead of 18 cm3/mol, because it was found that the thermodynamic basic volume unit of hydrogen-bonding solvents is consistently larger than the true molar volume.30,31 However, these differences (18 or 50) have little effect on the coexistence curves as far as the position and magnitude of the transition is concerned (up to 0.05 units of χ and 10% in magnitude).] First, we will examine the situation when all cross-links are of protein type, i.e., (∆νed)hg ) 0. In Figure 2, the swelling changes are displayed when passing from high to low to high degrees of swelling behavior, i.e., when the independent variable χj decreases. A discontinuous transition (abrupt increase in the degree of swelling) exists as a result of the unfolding of the folded domains. The magnitude of the jump in φ2 increases with an increasing fraction of folded domains in the gel. The fraction of folded domains is characterized by the weight ratio of the hydrogel polymer to the protein domain a ) whg/wf; that is, the width of the transition increases with decreasing a. At the same time, the transition is shifted to

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Figure 3. Swelling transition by unfolding of cross-links: effect of dilution at network formation given by φ0, the values are indicated; a ) 0.3, Nion ) 20, (νe)hg ) 0; other parameters as in Table 1.

Figure 4. Swelling transition by unfolding of cross-links: effect of the number of charges in unfolded network chain Nion ) 20, 50, 100, 200, and 300 from right to left; a ) 0.5, φ0 ) 0.3, (νe)hg ) 0; other parameters as in Table 1.

lower values of χj, because higher swelling pressure produced by the hydrogel polymer when its fraction decreases. The effect of dilution at network formation is shown in Figure 3. The magnitude of the transition as well as the critical value of χj increases with increasing φ0 because a less diluted gel produces swelling pressure (higher swelling force per crosslink). Increasing the fraction of ionized groups in the folded protein (Figure 4) does not change the position of the transition, because the ionized groups are exposed to the interactions only after unfolding. The magnitude of the transition increases with increasing product iNion as can be expected. The effect of additional cross-linking of the hydrophilic gel by conventional chemical cross-links is shown in Figure 5. With increasing cross-link density, the magnitude of the transition hardly changes but the transition is shifted to lower values of χj. This can be explained by the lower value of the force per cross-link f /rel because of an increasing number of cross-links.

Dusˇ ek et al.

Figure 5. Swelling transition by unfolding of cross-links: effect of contribution by tetrafunctional chemical cross-links (CX in Figure 1) (νe)hg indicated; a ) 0.5, φ0 ) 0.3, Nion ) 20; other parameters as in Table 1.

Figure 6. Reversible swelling transition by unfolding and folding of cross-links; allowed directions of changes indicated, for more explanation see text; a ) 0.5, φ0 ) 0.3, Nion ) 20, (νe)hg ) 0; other parameters as in Table 1.

Reversibility of the swelling changes is an important phenomenon. So far, there is little experimental evidence about the swelling changes when the direction of change of the parameter χj or the associated external parameter like temperature is reversed at a state when xf ) 0. Two options have been investigated: (a) the random coil of the protein / is reached motif folds again when the condition f /rel < f rel,crit (then xf changes from 0 to 1), and (b) xf remains equal to 0, / is positive or irrespective of the whether f /rel - f rel,crit negative. These situations are shown in Figure 6 for a typical set of parameters. Starting at some point corresponding, for instance, to χj ) 0.7, φ2 decreases along the curve D f A. At χj(A), a jumpwise transition occurs from φ2(A) to φ2(B); by further decreasing χj, φ2 changes from B to B′. When the direction of change of χj is reversed at B′, the decrease in swelling follows the curve B′ f B. At point B, no transition is observed but the swelling curve follows the curve from B / to C; xf ) 0 keeps holding and f /rel - f rel,crit > 0. At C, the


Figure 7. Reversible swelling transition by unfolding and folding of cross-links; allowed directions of changes indicated, for more explanation see text; a ) 0.5, φ0 ) 0.1, Nion ) 300, (νe)hg ) 0; other parameters as in Table 1. / condition f /rel - f rel,crit < 0 at xf ) 0 is met for the first time. Two options are possible: (a) a jumpwise transition C f D by which the folded structure of the protein motif is reformed and xf changes from 0 to 1; (b) xf remains equal to 0 and the degree of swelling further decreases along the curve C f E′. Thus, for option a the C f E′ part is absent, whereas for case b (irreversibility) the transition C f D is absent. For case b (transition to folded state is possible), the cycle D f A f B f C f D is possible, and full reversibility only exists for the part B h C. Starting somewhere at a point between D and A, an expansion transition is followed by a collapse transition; starting at conditions corresponding to χj between points B and C, a collapse transition can be followed by an expansion transition. An interesting situation is predicted when φ0 is low (dilution at network formation is high) and the contribution of the charged groups by the protein motif is also high. Then, two transitions, collapse and expansion ones, are predicted to occur irrespective of whether unfolding is reversible or not. However, their magnitudes are different (Figures 7 and 8). The swelling cycle (starting from the folded state at D′), φ2, decreases along the D f A curve, which is followed by A f B expansion transition. This transition is followed by weak and reversible change in the degree of swelling over a considerable range of χj values B h C. At C, a small increase in χj induces a large collapse transition. If the unfoldingfolding transition is reversible, the cycle is closed by passage from C to D. If this transition is not reversible and only part C h E is reversible, the swelling changes can take place reversibly along the paths E′ h E h C h B h B′. The C h E part is the classical collapse transition determined by balance between attractive and repulsive forces (brought in mainly by ionic effect; cf. refs 10-12). This transition occurring still at xf ) 1 induces the folding / < 0 by which the transition characterized by f /rel - f rel,crit ionic charges are again hidden in the folded structure when xf increases again to 1. One could think of a gel engine (converting chemical into mechanical energy, or a solution


Figure 8. Reversible swelling transition by unfolding and folding of cross-links; allowed directions of changes indicated, for more explanation see text; a ) 0.5, φ0 ) 0.1, Nion ) 500, (νe)hg ) 0; other parameters as in Table 1.

Figure 9. Same system as in Figure 8, but instead of affine network behavior [front factors Af ) 1, B ) 1, phantom network behavior assumed (Af ) fe/(fe - 2), B ) 0)].

concentrator) working, for instance, between different solvent activities of solution in the liquid phase in equilibrium, or between temperatures TA)B and TC)D. Alternatively, conversion of mechanical energy in thermal energy should also be possible. Externally imposed cyclic changes in swelling pressure (compression and decompression) may work as a heat pump (gel refrigerator). In the examples analyzed above, we have assumed the affine behavior of the network A ) 1, B ) 1 although the phantom network behavior A ) (fe - 2)/fe, B ) 0 is more characteristic of the swollen states. A comparison of Figures 8 and 9 shows that the difference in transitions is small; the application of the phantom network model makes the transition somewhat more pronounced. Conclusions The theoretical analysis has shown that a transition in hybrid biopolymer networks with folded motifs induced by increasing osmotic pressure produced by the hydrogel networks can occur in a realistic range of structural parameters. In a certain range of external and structural parameters, this unfolding transition can be coupled with the conventional

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collapse transition occurring in disordered gels as a result of interplay between attractive and repulsive (mainly ionic) interactions. If the unfolding-folding process is reversible, two states are expected to exist at the same values of external parameters depending on the direction of change of the osmotic pressure. This is due to the existence of different interacting sites corresponding to folded or unfolded states of cross-links. Acknowledgment. Partial financial support by the Academy of Sciences of the Czech Republic (Project AVOZ 4050913) and by the Grant Agency of the Charles University, Czech Republic (Project 166/2001/B) is gratefully acknowledged. References and Notes (1) Lee, J.; Macosko, C. W.; Urry, D. W. Macromolecules 2001, 34, 5968-5974. (2) Di Zio, K.; Tirrell, D. A. Macromolecules 2003, 36, 1553-1558. (3) Kakugo, A.; Sugimoto, S.; Gomg, J. P.; Osada, Y. AdV. Mater. 2002, 14, 1124-1126. (4) Chen, L.; Kopecˇek, J.; Stewart, R. J. Bioconjugate Chem. 2000, 11, 734-740. (5) Wang, C.; Stewart, R. J.; Kopecˇek, J. Nature 1999, 397, 417-420. (6) Ulbrich, K.; Strohalm, J.; Kopecˇek, J. Biomaterials 1982, 3, 150154. (7) te Nijenhuis, K. AdV. Polym. Sci. 1997, 130, 1-252. (8) Dusˇek, K.; Patterson, D. J. Polym. Sci., Part A-2 1968, 6, 12091216. (9) Tanaka, T. Phys. ReV. Lett. 1978, 40, 820-823. (10) ResponsiVe Gels. Volume Phase Transition, Vol. I; Dusˇek K., Ed.; AdV. Polym. Sci.; Springer: New York, 1993, Vol. 109. (11) ResponsiVe Gels. Volume Phase Transition, Vol. II; Dusˇek, K., Ed.; AdV. Polym. Sci.; Springer: New York, 1993, Vol. 110.

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BM034219S

Swelling Pressure Induced Phase-Volume Transition in Hybrid

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