Effect of Ligand Binding on Polymer Diffusiophoresis

: Diffusiophoresis is the migration of a macromolecule in response to a concentration gradient of a cosolute in liquids. Diffusiophoresis of polyethylene glycol (PEG) in water occurs from high to low concentration of trimethylamine-N-oxide (TMAO). This is consistent with the preferential hydration of PEG observed in the presence of TMAO. In other words, PEG migrates in the direction in which it lowers its chemical potential. On the other hand, although PEG is found to preferentially bind urea in water, PEG diffusiophoresis still occurs from high to low urea concentration. Thus, PEG migrates in the direction that increases its chemical potential in the urea case. Here, a ligand-binding model for polymer diffusiophoresis in the presence of a cosolute that preferentially binds to polymer is developed. It includes both actual polymer–ligand binding and the effect of the polymer on cosolute thermodynamic activity. This model shows that polymer–cosolute binding has a marginal effect on polymer diffusiophoresis and indicates that weak repulsive interactions, such as hard-core exclusion forces, are the main factor responsible for the observed PEG diffusiophoresis from high to low urea concentration. This work contributes to a better understanding of diffusiophoresis of macromolecules in response to gradients of nonelectrolytes.


Introduction
Macromolecules and colloidal particles in general are subject to diffusion-based transport in multi-component liquids relevant to a broad range of biochemical, biotechnological, geochemical, and industrial processes, such as centrifugation [1], dialysis [2], adsorption [3], crystallization [4], transport in microfluidics [5], living systems [6], gel media [7], insertion into dead-end pores [8], controlled release [9], and reaction kinetics [10,11].Since concentration gradients of mixture components are usually present in these systems, diffusion coefficients are important for modeling the kinetic evolution of spatial concentration profiles within the liquid mixture.
A transport phenomenon known as diffusiophoresis has attracted much attention in recent years [12,13].This is the isothermal and isobaric migration of a particle through a liquid in response to the directional concentration gradient of another solute (cosolute) present in the same liquid mixture.Although diffusiophoresis may occur in any liquid, this transport phenomenon is normally considered in the context of aqueous mixtures.Here, the solute responsible for diffusiophoresis of a colloidal particle is typically an inorganic salt or a low-molecular-weight water-soluble organic molecule [14,15].
Most studies have focused on diffusiophoresis of charged colloidal particles in the presence of salt gradients [16][17][18][19][20].In this case, diffusiophoresis is caused by an electrophoretic mechanism that is ultimately related to the difference in mobility between the two salt ions in water (diffusion potential).However, it has been experimentally shown that diffusiophoresis is also observed for the macromolecule in polyethylene glycol (PEG) and PEG-based micelles [21][22][23].This implies that gradients of salts or osmolytes can induce the migration of colloidal particles that are electrically neural.
PEG is a nonionic hydrophilic polymer found in many aqueous formulations relevant to pharmaceutical and biotechnological applications [24].Furthermore, an important class of water-soluble colloidal particles is represented by neutral particles whose interfacial properties are modified or governed by PEG.Indeed, this polymer has been employed for coating the surface of inorganic nanoparticles [25,26], proteins [27], micelles [28], and vesicles [29,30].Since diffusiophoresis is mostly an interfacial phenomenon, understanding the diffusiophoresis of PEG coils is of fundamental importance for understanding the diffusiophoresis of PEG-based colloidal particles.
In previous studies, PEG diffusiophoresis has been connected to a thermodynamic phenomenon known as preferential hydration.This is described by the thermodynamic excess of water molecules near a macromolecule compared to bulk [31][32][33].PEG is preferentially hydrated in the presence of osmolytes, such as trimethylamine-N-oxide (TMAO), and salts, such as NaCl and Na 2 SO 4 [21][22][23]34].Since PEG is preferentially hydrated in the presence of these cosolutes, diffusiophoresis occurs from a high to low cosolute concentration (positive diffusiophoresis) in order to lower PEG chemical potential.
Diffusiophoresis and preferential hydration can be examined within the framework of multicomponent diffusion [35,36].Specifically, diffusion in a ternary aqueous system is characterized by a 2 × 2 diffusion coefficient matrix, in which one of the two cross-term diffusion coefficients relates to macromolecule diffusiophoresis due to the cosolute concentration gradient.The other cross-term, which is denoted as cosolute osmotic diffusion, describes cosolute diffusion due to a macromolecule concentration gradient and is closely related to preferential hydration [22].Specifically, it has been shown that cosolute osmotic diffusion is important for determining water thermodynamic excess and, therefore, it characterizes the preferential hydration component of PEG diffusiophoresis [22,23].
The behavior of PEG diffusiophoresis was examined by considering a local domain model, in which cosolute concentration is depleted near the particle (local domain) compared to bulk fluid [22,31].In the presence of particle diffusiophoresis, the local domain may be split into an inner local domain, in which fluid molecules move together with the particle, and an outer local domain, enclosing molecules moving with bulk fluid.It is only the outer domain that is responsible for diffusiophoresis.According to this model, a solute or solvent binding to a particle contributes to the inner fraction domain, whereas interactions that are weak compared to thermal energy or excluded-volume interactions [33,37,38] should contribute to the outer fraction domain.The fraction of the inner domain can be determined by combining diffusiophoresis and osmotic diffusion data.
There are also cosolutes that preferentially bind PEG: urea or thiocyanate salts [22,39].In the presence of these cosolutes, water is depleted near PEG compared to bulk, leading to negative values of water thermodynamic excess.In this case, there is an expectation that diffusiophoresis occurs from a low to high cosolute concentration (negative diffusiophoresis) in order to lower PEG chemical potential.In contrast, it has been experimentally found that PEG diffusiophoresis remains positive even in the presence of such cosolutes [22,39].In this paper, a model explaining the occurrence of positive diffusiophoresis even in the presence of a cosolute preferentially binding is developed.It specifically applies to the case of both the polymer and cosolute being electrically neutral and is suitable to describe PEG diffusiophoresis in the presence of urea.

Discussion
This section is organized into the following three subsections.In Section 2.1, the theoretical background on macromolecule diffusiophoresis and cosolute osmotic diffusion coefficients is outlined.These two transport coefficients are introduced within the framework of multicomponent-diffusion in ternary liquid mixtures.In Section 2.2, experimental data on PEG diffusiophoresis and cosolute osmotic diffusion in the presence of TMAO (preferential hydration) and urea (preferential binding) will be reviewed and interpreted by employing the local domain model.In Section 2.3, a novel ligand-binding model is developed in order to explain the positive diffusiophoresis of a polymer in the presence of cosolutes that preferentially bind to it.

Theoretical Background
For a ternary macromolecule(P)-cosolute(S)-solvent(W) system in isothermal and isobaric conditions and in the absence of convection, we can write the following [22]: where J P and J S are the molar fluxes of macromolecule (P) and cosolute (S) in the solventfixed reference frame, respectively, C P and C S are the corresponding molar concentrations, and the four D ij 's (with i,j = P,S) are multicomponent diffusion coefficients in the solventfixed reference frame.The main-term coefficients, D PP and D SS , describe the flux of the macromolecule and cosolute due to their own concentration gradients, while the cross-term coefficients, D PS and D SP , describe the flux of a solute due to the concentration gradient of the other solute.The cross-term, D PS , corresponds to cosolute-induced macromolecule diffusiophoresis, while the other cross-term, D SP , describes the cosolute osmotic diffusion due to the macromolecule concentration gradient.
In the limit of C P →0, the diffusion rate of a macromolecule, v P = lim CP→0 (J P /C P ), is described by [40], as follows: where the second term in Equation ( 2) describes macromolecule diffusiophoresis while the first term is the Brownian diffusion rate caused by the entropic restoring force, with D P ≡ lim CP→0 D PP being the Brownian mobility (diffusion coefficient) of the macromolecule.In Equation ( 2), the unitless coefficient, DPS , is a reduced diffusiophoresis coefficient describing the effect of cosolute through the thermodynamic force, −∇µ S , where µ S is the cosolute chemical potential, R the ideal gas constant and T the absolute temperature.We have the following [36]: where ν S = 1 for non-electrolytes (e.g., urea or TMAO) and ν S = 2 for symmetric electrolytes, and y S is a known [22] thermodynamic factor characterizing the thermodynamic nonideality of the binary cosolute-water system.Non-equilibrium thermodynamics can be then used to show that DPS is connected to the cross-term, D PS in Equation (1a), by [22,35], as follows: Non-equilibrium thermodynamics also show the following [22,35]: where γ ≡ lim CP→0 (∂µ P /∂µ S ) CP is a thermodynamic preferential interaction coefficient, with µ P being the macromolecule chemical potential.The other coefficient, λ, is a unitless Onsager transport coefficient defined by rewriting Equation (2) in the following way [21]: Note that the negative sign preceding λ is a convention, chosen to ensure that this parameter assumes positive values in preferential hydration conditions.
Cosolute osmotic diffusion can be described by the ratio between the cross-term and main-term cosolute diffusion coefficient [35].This is a unitless coefficient, DSP , given by the following [22]: where C SP ≡ −lim CP→0 (∂C S /∂C P )µ S is a cosolute partitioning coefficient [41] and α ≡ D P /D S is a macromolecule-to-cosolute mobility ratio, with D S being the cosolute diffusion coefficient in the solvent-fixed reference frame, with α ≈ 0.1 or smaller.Since the αλ term in Equation ( 7) is small compared to the partitioning coefficient, C SP , we have DSP ≈ C SP .This means that DSP is approximately a thermodynamic quantity.The thermodynamic coefficient, γ, is linked to C SP by [21], as follows: where V P ≡ V P − (ν S y S ) −1 V S with V P and V S being macromolecule and cosolute partial molar volumes, respectively.Note that V P = V P is an excellent approximation because The coefficients, DPS and DSP , are extracted from ternary diffusion coefficients using Equations ( 4) and ( 7) as a function of cosolute concentration, C S .Although experimental values of DSP (C S ) alone can be directly used to extract γ(C S ) approximately, a combination of DPS (C S ) and DSP (C S ) data can be used to rigorously extract γ(C S ) and γ(C S ) from combination of Equations ( 5), ( 7) and (8).
As previously mentioned, water thermodynamic excess, ν W , is introduced by employing the local domain model [22,31].The following can be shown [21,42]: where V W is water's partial molar volume.Thus, the value of ν W is extracted by fitting the experimental data of γ(C S ) to Equation (9).We then assume that the local domain is split into an inner local domain (I), in which fluid molecules move together with particle, and an outer local domain (II), enclosing molecules moving with bulk fluid.The thermodynamic excesses in the inner and outer domain are ν W (I) and ν W (II) , respectively; with [22].According to this model, the ratios, λ/γ and DPS /γ, are constants representing the two complementary fraction of inner and outer domain, respectively.We have the following: Thus, the values of ν W (I) and ν W (II) can be extracted from the experimental data of ν W and DPS /γ.

Analysis of Experimental Data
In this section, the DPS (C S ) and DSP (C S ) data [22] on aqueous PEG (nominal molecular weight, 20 kg/mol) in the presence of TMAO and urea at 25 • C will be reviewed.These data will be then used to extract the corresponding values of ν W (I) and ν W (II) .It is convenient to examine cosolute osmotic diffusion prior to PEG diffusiophoresis due to its more direct connection to the thermodynamic quantity, γ.In Figure 1, cosolute osmotic diffusion coefficient, DSP , as a function of osmolyte concentration, C S , is shown.As we can see, DSP data are positive for both cosolutes, thereby implying that the cosolute diffuses from high to low PEG concentration.We can also see that DSP lin-early increases with C S starting from DSP (0) = 0.This behavior is in agreement with DSP ≈ C SP = (V P + ν W V W )C S based on Equations ( 8) and ( 9), with V W = 18.07 cm 3 •mol −1 .For comparison, the reference line, V P C S (with V P = 16.7 dm 3 •mol −1 ) is included in the same figure.Positive deviations from this line correspond to positive values of water thermodynamic excess, ν W .We can then deduce that PEG is preferentially hydrated in the presence of TMAO.Indeed, the application of Equations ( 5), ( 7) and (8) to DPS (C S ) and DSP (C S ) data, which confirms that DSP ≈ C SP is a good approximation, rigorously yields γ(C S ).We can then determine the thermodynamic excess of water molecules [43] near PEG using Equation ( 9), which is ν W = 2600 (5.7 per ethoxy group) in the TMAO case.In contrast, urea is preferentially binding to PEG because DSP data lay below the reference line.In this case, the water thermodynamic excess is ν W = −580 (−1.3 per ethoxy group).It is interesting to observe that the effect of TMAO and urea on PEG follows the same behaviour observed in the case of proteins.
Thus, the values of νW (I) and νW (II) can be extracted from the experimental data of νW and D ̂PS/γ.

Analysis of Experimental Data
In this section, the D ̂PS(CS) and D ̂SP(CS) data [22] on aqueous PEG (nominal molecular weight, 20 kg/mol) in the presence of TMAO and urea at 25 °C will be reviewed.These data will be then used to extract the corresponding values of νW (I) and νW (II) .
It is convenient to examine cosolute osmotic diffusion prior to PEG diffusiophoresis due to its more direct connection to the thermodynamic quantity, γ.In Figure 1, cosolute osmotic diffusion coefficient, D ̂SP, as a function of osmolyte concentration, CS, is shown.As we can see, D ̂SP data are positive for both cosolutes, thereby implying that the cosolute diffuses from high to low PEG concentration.We can also see that D ̂SP linearly increases with CS starting from D ̂SP (0) = 0.This behavior is in agreement with D ̂SP ≈ CSP = (V̅ P + νWV̅ W)CS based on Equations ( 8) and ( 9), with V̅ W = 18.07 cm 3 •mol −1 .For comparison, the reference line, V̅ PCS (with V̅ P = 16.7 dm 3 •mol −1 ) is included in the same figure.Positive deviations from this line correspond to positive values of water thermodynamic excess, νW.We can then deduce that PEG is preferentially hydrated in the presence of TMAO.Indeed, the application of Equations ( 5), ( 7) and ( 8) to D ̂PS(CS) and D ̂SP(CS) data, which confirms that D ̂SP ≈ CSP is a good approximation, rigorously yields γ(CS).We can then determine the thermodynamic excess of water molecules [43] near PEG using Equation ( 9), which is νW = 2600 (5.7 per ethoxy group) in the TMAO case.In contrast, urea is preferentially binding to PEG because D ̂SP data lay below the reference line.In this case, the water thermodynamic excess is νW = −580 (−1.3 per ethoxy group).It is interesting to observe that the effect of TMAO and urea on PEG follows the same behaviour observed in the case of proteins.In Figure 2, the PEG diffusiophoresis coefficient, D ̂PS, as a function of cosolute concentration, CS, is shown.As we can see, D ̂PS data are also positive, thereby implying that PEG diffusiophoresis occurs from a high to low osmolyte concentration in both cases.Moreover, as in the case of osmotic diffusion data, D ̂PS linearly increases with CS starting from D ̂PS(0) = 0.The slope trend of diffusiophoresis data is the same as that of osmotic In Figure 2, the PEG diffusiophoresis coefficient, DPS , as a function of cosolute concentration, C S , is shown.As we can see, DPS data are also positive, thereby implying that PEG diffusiophoresis occurs from a high to low osmolyte concentration in both cases.Moreover, as in the case of osmotic diffusion data, DPS linearly increases with C S starting from DPS (0) = 0.The slope trend of diffusiophoresis data is the same as that of osmotic diffusion data in Figure 1, with PEG diffusiophoresis being larger in the TMAO case.In the urea case, although PEG prefers to interact more with urea than water, PEG diffusiophoresis still occurs from the high to low urea concentration.
Liquids 2023, 3, FOR PEER REVIEW 6 diffusion data in Figure 1, with PEG diffusiophoresis being larger in the TMAO case.In the urea case, although PEG prefers to interact more with urea than water, PEG diffusiophoresis still occurs from the high to low urea concentration.Since both D ̂PS and γ are directly proportional to CS, the ratio, D ̂PS/γ, is a constant, consistent with the local domain model.In the TMAO case, we have D ̂PS/γ = 0.14, which gives νW (I) = +2200 (4.8 per ethoxy group) in the inner domain and νW (II) = +360 (0.80 per ethoxy group) in the outer domain.This result is consistent with a weakening of PEG preferential hydration as we move from the inner to the outer domain.In the urea case, Since both DPS and γ are directly proportional to C S , the ratio, DPS /γ, is a constant, consistent with the local domain model.In the TMAO case, we have DPS /γ = 0.14, which gives ν W (I) = +2200 (4.8 per ethoxy group) in the inner domain and ν W (II) = +360 (0.80 per ethoxy group) in the outer domain.This result is consistent with a weakening of PEG preferential hydration as we move from the inner to the outer domain.In the urea case, the negative value of DPS /γ = −0.10 is the result of DPS being positive and γ being negative (due to urea preferential binding, ν W = −580).This leads to ν W (I) = −640 (−1.4 per ethoxy group) for the inner domain but ν W (II) = +60 (0.13 per ethoxy group) for the outer domain.The negative value of ν W (I) is attributed to an actual PEG-urea binding process in the inner domain.However, a simple weakening of PEG-urea binding interaction should yield a negative value of ν W (II) in the outer domain too.Thus, the observed sign inversion of water thermodynamic excess is inconsistent with an actual physical division into inner and outer spatial domains.We should, therefore, generalize the actual definition of inner and outer domains as just two categories separating strong binding interactions from non-binding interactions, independent of spatial location.For example, excluded volume interactions invariably start at the macromolecule surface, but they should be formally attributed to the outer domain [33,37,38].We can, therefore, explain PEG diffusiophoresis in the urea case by considering that the overall PEG-urea interactions are the result of an actual PEG-urea binding process (inner domain, ν W (I) < 0) and residual excluded volume interactions, which are associated with the outer domain and are responsible for ν W (II) > 0. A ligand-binding model supporting this conclusion is developed in Section 2.3.

Ligand-Binding Model for Polymer Diffusiophoresis
In this section, the mathematical expressions of γ and DPS are derived by considering both polymer-ligand-binding and non-binding interactions.It will be shown that this model is consistent with experimental results on PEG diffusiophoresis in the presence of urea.
We start by assuming that there is a reversible binding of cosolute ligand S on the linear chain of a polymer, P, according to the following: with n being the total number of sites on the polymer chain.This binding process is characterized by the chemical equilibrium condition as follows: where a S is the thermodynamic activity of cosolute, S. Note that the activity coefficient of the PS i−1 complex is assumed to be the same as that of the PS i complex.In this way, the ratio of their activities is the same as that of their concentrations, Even if these activity coefficients are not assumed to be equal to each other, they will approach one when taking the limit of C P →0, consistent with the application of Equation ( 2).We then assume that the thermodynamic activity of cosolute can be written as follows: where with [S] being the molar concentration of free cosolute in the polymer-cosolute-water mixture.The cosolute concentration, [S] , describes the effect of C P on cosolute thermodynamic activity.It represents the concentration of free S in a binary cosolute-water reservoir that is in chemical equilibrium with the ternary polymer-cosolute-water mixture through a membrane that is not permeable to polymer.In Equation (13), ε is a unitless interaction coefficient characterizing the effect of polymer on [S] to the first-order with respect to C P .If ε is positive, then [S] is larger than [S].This situation describes the excluded volume effect of the polymer chain exerted on cosolute.Higher-order terms in the denominator of Equation ( 13) are not included because they vanish in the limit of C P →0.In Equation ( 12), f S is the activity coefficient of L. It is assumed to be the same as that of the binary cosolute-water reservoir.This implies that f S is a function of [S] .The thermodynamic non-ideality factor, y S in Equation ( 2), is then given by the following: The cosolute partitioning coefficient, C SP , can written as follows: The partial derivatives of lna S in the numerator and denominator of Equation ( 15) are given by the following: where we have used Equation ( 13) to deduce the following: After substituting Equations (16a) and (16b) into Equation ( 15), we have the following: Note that C SP does not depend on the non-ideality thermodynamic factor, y S .To determine C S ([S], C P ), we consider the mass balances for P and S as follows: where we have also used Equation (11) and introduced the cumulative associative constant We can remove polymer free concentration, [P], from Equation (18b) by combining the two mass balances in the following way: Liquids 2023, 3

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We are now in a position to differentiate C S with respect to [S] and C P .This yields the following: where we have introduced the mean degree of cosolute binding, as follows: Note that < i > = ∑ n i=0 i [PS i ]/C P (case of j = 1) in Equation (21b) represents the average number of cosolute molecules bound to a polymer chain, while the factor, < i 2 > − < i > 2 , in Equations (21a) and (21b) represents the variance of the number of cosolute molecules bound to a polymer chain.Since the terms containing the variance factor in Equations (21a) and (21b) are directly proportional to C P , they can be neglected in the limit of C P →0.Thus, substitution of (∂C S /∂[S]) C P = 1 and (∂C S /∂C P ) [S] = < i > 0 into Equation ( 18), with <i> 0 ≡ lim CP→0 <i>, finally yields: where we have also used C S = lim CP→0 [S].To appreciate the dependence of <i> 0 on C S , we can consider the special case of identical and independent binding sites, [44] which corresponds to β i ≡ n i K −i in Equation ( 20), with K being the intrinsic equilibrium dissociation constant of a site.In this case, the average number of cosolute molecules bound to a polymer chain is <i> 0 = nC S /(K + C S ).Due to cosolute binding, <i> 0 is directly proportional to C S at low cosolute concentration and approaches the value of the total number of sites, n = lim CS→∞ <i> 0 at a high cosolute concentration.
To obtain the expression of DPS , we need to derive the expression of γ from C SP .Accordingly, we use Equation (8) to write the following: where Ω ≡ ε V P − V P .This volumetric coefficient is positive in the presence of excludedvolume interactions as it will be discussed below.Note that 1 − C S V S ≈ 1 because C S V S represents a relatively small volume fraction in the experimental urea concentration range shown in Figures 1 and 2. If we also assume that <i> 0 = (n/K) C S (with K >> C S ), Equation (24) reduces as follows: We now turn our attention to the transport coefficients, λ, introduced by Equation ( 6).We introduce a simple diffusion law for each PS i complex in a dilute solution of polymer, as follows: where J i , D PS i , and µ i are the molar flux, tracer diffusion coefficient, and chemical potential of the PS i complex i, respectively.Note that diffusion of the free polymer is described by the case of i = 0.The expressions of total fluxes of polymer (J P ) and cosolute (J S ) follow the mass balances given by Equations (19a) and (19b).We, therefore, have the following: all polymer diffusiophoresis is also predicted to be relatively small compared to that of polymer-cosolute non-binding interactions.The observed PEG diffusiophoresis from high to low urea concentration (positive diffusiophoresis) is attributed to repulsive hard-core exclusion interactions.
Funding: This research was funded by Texas Christian University RCAF grant number 61008.

Figure 1 .
Figure 1.Cosolute osmotic diffusion coefficient, D ̂SP, as a function of cosolute concentration, CS, in the TMAO and urea cases.Dashed line represents V̅ PCS.Graphic representation on the right describes cosolute osmotic diffusion occurring from high to low concentration of PEG coils.Data taken from Ref. [22].

Figure 1 .
Figure 1.Cosolute osmotic diffusion coefficient, DSP , as a function of cosolute concentration, C S , in the TMAO and urea cases.Dashed line represents V P C S .Graphic representation on the right describes cosolute osmotic diffusion occurring from high to low concentration of PEG coils.Data taken from Ref. [22].

Figure 2 .
Figure 2. PEG diffusiophoresis coefficient, D ̂PS, as a function of cosolute concentration, CS, in the TMAO and urea cases.Inset is a graphic representation of PEG diffusiophoresis occurring from high to low cosolute concentration.Data taken from Ref. [22].

Figure 2 .
Figure 2. PEG diffusiophoresis coefficient, DPS , as a function of cosolute concentration, C S , in the TMAO and urea cases.Inset is a graphic representation of PEG diffusiophoresis occurring from high to low cosolute concentration.Data taken from Ref. [22].