Establishment of a Physical Model for Solute Diffusion in Hydrogel ...

Article pubs.acs.org/JPCB

Establishment of a Physical Model for Solute Diffusion in Hydrogel: Understanding the Diffusion of Proteins in Poly(sulfobetaine methacrylate) Hydrogel Yuhang Zhou,† Junjie Li,*,§ Ying Zhang,† Dianyu Dong,† Ershuai Zhang,† Feng Ji,† Zhihui Qin,† Jun Yang,∥ and Fanglian Yao*,†,‡ †

School of Chemical Engineering and Technology and ‡Key Laboratory of Systems Bioengineering, Ministry of Education, Tianjin University, Tianjin 300072, China § Department of Advanced Interdisciplinary Studies, Institute of Basic Medical Sciences and Tissue Engineering Research Center, Academy of Military Medical Sciences, Beijing 100850, China ∥ The Key Laboratory of Bioactive Materials, Ministry of Education, College of Life Science, Nankai University, Tianjin 300071, China ABSTRACT: Prediction of the diffusion coefficient of solute, especially bioactive molecules, in hydrogel is significant in the biomedical field. Considering the randomness of solute movement in a hydrogel network, a physical diffusion RMP-1 model based on obstruction theory was established in this study. The physical properties of the solute and the polymer chain and their interactions were introduced into this model. Furthermore, models RMP-2 and RMP-3 were established to understand and predict the diffusion behaviors of proteins in hydrogel. In addition, zwitterionic poly(sulfobetaine methacrylate) (PSBMA) hydrogels with wide range and fine adjustable mesh sizes were prepared and used as efficient experimental platforms for model validation. The Flory characteristic ratios, Flory−Huggins parameter, mesh size, and polymer chain radii of PSBMA hydrogels were determined. The diffusion coefficients of the proteins (bovine serum albumin, immunoglobulin G, and lysozyme) in PSBMA hydrogels were studied by the fluorescence recovery after photobleaching technique. The measured diffusion coefficients were compared with the predictions of obstruction models, and it was found that our model presented an excellent predictive ability. Furthermore, the assessment of our model revealed that protein diffusion in PSBMA hydrogel would be affected by the physical properties of the protein and the PSBMA network. It was also confirmed that the diffusion behaviors of protein in zwitterionic hydrogels can be adjusted by changing the cross-linking density of the hydrogel and the ionic strength of the swelling medium. Our model is expected to possess accurate predictive ability for the diffusion coefficient of solute in hydrogel, which will be widely used in the biomedical field.

1. INTRODUCTION Hydrogels are often regarded as biomaterials because of their ability to contain large amounts of water and display characteristics similar to those of natural tissues.1,2 Furthermore, their three-dimensional (3-D) network structure is extremely suitable for accommodating bioactive compounds, such as therapeutic proteins and peptides3 and even live cells,4 which renders them useful for protein delivery, protein separation, therapeutic implant, and cell encapsulation.5−7 Almost all of these applications involve protein diffusion, whose behavior in hydrogel will directly affect the efficiency of the applications. Taking cell encapsulation as an example, to maintain the excellent cell viability, it must be ensured that the proteins (nutrients and metabolic waste products) can freely diffuse and their diffusion rate can be controlled in different stages of cell growth.8,9 Hence, the study of protein diffusion in hydrogel has attracted great attention. The researchers not only © 2017 American Chemical Society

focused on the experimental investigation of the diffusion process and its characteristics but also established a series of physical models to predict the diffusion coefficient.10−21 Because of their high hydration and ultralow fouling properties,22,23 zwitterionic hydrogels have recently been widely used in biomedical applications, such as drug-delivery carriers,24,25 in vivo implants, cell encapsulation matrixes, and so forth.26−30 Poly(carboxybetaine methacrylate) (PCBMA) hydrogels after modification with additional cell-adhesion moieties can provide an excellent 3-D environment for cell growth, which was a potential carrier for cell encapsulation.26−30 Ishihara et al. found that a cytocompatible phospholipid polymer hydrogel can be used as a 3-D cell Received: October 13, 2016 Revised: December 18, 2016 Published: January 6, 2017 800

DOI: 10.1021/acs.jpcb.6b10355 J. Phys. Chem. B 2017, 121, 800−814

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The Journal of Physical Chemistry B

accurately predict the diffusion coefficient of a solute in hydrogel. 2.1. Theoretical Background. Physical models based on obstruction theory assumed that the presence of impenetrable and stationary polymer chains in the hydrogel network leads to a longer path length of solute diffusion. According to this assumption, it is generally accepted that a solute molecule diffuses randomly in the hydrogel across a succession of the polymer mesh with appropriate size. Therefore, the polymer chains play a role of sieving the solute that can only pass through the mesh between the adjacent polymer chains.33,34 The physical model based on obstruction theory was established mainly to deduce an appropriate sieving factor, which was governed by the probability of the solute to find a series of meshes that allow it to pass through. Ogston et al. believed that solute diffusion in the hydrogel occurs by a succession of directionally random unit steps and that the unit step does not take place if the solute encounters a polymer chain. The unit step equals the root-mean-square average diameter of spherical spaces existing between the polymer networks.36 The Ogston model is expressed in eq 1A

encapsulation culture matrix, and the proliferation of cells encapsulated in this hydrogel can be modulated by the concentrations of the polymers and the swelling degree of the hydrogel.27 A further understanding of the diffusion behaviors of proteins or other solutes in these zwitterionic hydrogels will provide a clearer guidance for their applications in the biomedical field. Different from other hydrophilic polymer hydrogels, the zwitterionic hydrogels are characterized by their antipolyelectrolyte behavior. For example, the zwitterionic polymer chain presents a shrinkage state in water but stretches in an aqueous solution in the presence of salt ions, which provides the possibility to adjust the mesh size of the hydrogel and the flexibility of the polymer chain by changing the ionic strength of the swelling medium. In addition, many zwitterionic hydrogels are also temperature-sensitive (e.g., poly(sulfobetaine methacrylate), PSBMA) and/or pH-sensitive (e.g., PCBMA). The network structures and properties of the zwitterionic hydrogels can be easily adjusted by these stimulus responses.31,32 As a result, zwitterionic hydrogels are not only excellent biomaterials but also promising matrixes for diffusion adjustment of the solute. However, to the best of our knowledge, there are only few systematic studies on the protein diffusion in zwitterionic hydrogels, despite the broad applications of the zwitterionic hydrogel in the biomedical field and the importance of the diffusion behavior and diffusion adjustment of protein in zwitterionic hydrogels. In this work, we established a new physical model based on obstruction theory to predict solute diffusions in hydrogels, trying to understand the relationship between the diffusion behaviors of solutes in hydrogels and their structures/properties thereby providing guidance for the structural design of hydrogels in the biomedical field. Specifically, zwitterionic PSBMA hydrogels were selected as the model diffusion matrixes. A wide range of mesh sizes of the PSBMA hydrogels were realized by changing the cross-linking density, and fine adjustable mesh sizes were obtained by controlling the ionic strength of the swelling medium. The network characteristics and related parameters of the PSBMA hydrogels were investigated. Bovine serum albumin (BSA), immunoglobulin G (IgG), and lysozyme (LYZ) with different sizes, isoelectric points, and flexibilities were used as model proteins. The diffusion coefficients of the proteins in PSBMA hydrogels were determined by fluorescence recovery after photobleaching (FRAP) measurements, and the results were used for model validation. Furthermore, the model predictions and experimental values were compared, attempting to reveal how the properties of the protein and the PSBMA network and the interaction between them affect the diffusion behaviors of the protein in the PSBMA hydrogel.

⎤ ⎡ (r + rf ) D v2,s ⎥ = exp⎢ − s D0 rf ⎦ ⎣

(1A)

where D0 and D are the solute diffusion coefficients in the aqueous solution and hydrogel, respectively, rs and rf are the radii of the solute and polymer chain, respectively, and v2,s is the polymer volume fraction of the hydrogel at the equilibrium swelling state. Yusuda et al. held the opinion that the sieving factor equaled the percentage of the number of meshes whose area was larger than the solute effective cross-sectional area to the total number of meshes.37 However, Lustig and Peppas believed that some solute molecules had the same effective cross-sectional area but might have different hydrodynamic radii. Sieving factor is a function of the proportion of the hydrodynamic radius of the solute and the average mesh size of the hydrogel.18 On the basis of the viewpoint of Lustig and Peppas, Amsden also believed that the sieving factor should be the percentage of the number of meshes larger than the solute hydrodynamic size to the total number of meshes. The Amsden diffusion model was obtained by combining the mesh size distribution function raised by Ogston. The expression of this model is shown in eq 1B,14 where P is the sieving factor and r ̅ is the average radius of the openings between the polymer chains P=

2. MODEL DEVELOPMENT Several physical models are used for solute diffusion in polymer hydrogel,33−35 many of which are based on obstruction theory.14−17,36−39 However, the physical properties of the solute or the polymer chain and the interactions between them had not been considered in most of the obstruction models. Therefore, the characteristics of a specific solute−hydrogel system cannot be reflected in these models. In addition, the naturally random movement of the solute in a hydrogel network should be comprehensively considered. Therefore, in this section, we took into account the self-diffusion characteristics of the solute in the hydrogel network and established a new diffusion model based on obstruction theory, attempting to

2⎤ ⎡ D π ⎛ r + rf ⎞ ⎥ = exp⎢ − ⎜ s ⎟ ⎢⎣ 4 ⎝ r ̅ + rf ⎠ ⎥⎦ D0

(1B)

Therefore, obstruction diffusion models can be divided into two categories: models based on the relationship of effective area and models based on linear size. Regardless of whether the starting point is an area or a linear size, these models shared a common view. For a certain mesh, these models simply considered two states that the solute is able or not to be allowed to pass through the mesh. They did not take into account whether the probability this mesh contributed to the sieving factor had a stronger relationship with the size or area of the mesh. Because of the presence of a mesh size distribution in the hydrogel, some of the meshes are significantly larger than the solute. It can be assumed that the probability contributed by these meshes is higher than that of the smaller meshes. It 801


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Figure 1. Schematic diagrams of the conversion process of a 3-D network sieve solute to a 2-D mesh plane.

Figure 2. Schematic diagrams of the process of model establishment (A−C). Step 1 indicates that there is a distribution function of the hydrogel mesh size and only part of the meshes is able to diffuse for a solute of certain size. Step 2 shows that only when the center of the solute projects onto the light blue areas, the solute can pass through the mesh.

sieving factor equals the probability that the solute can pass through the mesh plane from any position in this plane. The diffusion system and its screening process have the following assumptions: (1) The hydrogel is composed of a long straight polymeric fiber network with a random orientation. (2) The distribution function of the mesh size can be expressed by the radius distribution function of the spherical space in the long straight polymer fiber network with random orientation, which was proposed by Ogston.40 (3) The solute is considered to be spherical, and its size is expressed by its hydrodynamic radius. (4) In the process of plane sieving, it is considered that the movement direction of the solute is perpendicular to the mesh plane. Although there are many other angles, the solute molecules are considered to be arbitrarily located in a plane that is a unit step away from the mesh plane. Thus, to some extent, other angles can be compensated by the different initial positions of the solute. This would fully consider the random nature of the solute movement. This issue could be explained in another way. When the solute is positioned in the hydrogel network, it is believed that the grid can be distributed on the spherical surface of the solute. Because the solute can move in any direction, any contact with the surface of the sphere can be regarded in a vertical direction. (5) When the mesh size distribution satisfies the Ogston distribution function, the hydrogel network is regularized to facilitate the calculation and analysis. It should be noted that the Ogston distribution function represents the size distribution of the spherical space between the polymer chains. After the transformation of the sieving process, there is no inherent difference between the circle and square in the calculation process of the model establishment. First, we temporarily do not consider the radius of the polymer chain, and focus on the mesh regions that can be used

can be also assumed that even when all of the meshes or areas adjacent to a solute in the hydrogel network are larger than the solute itself, the solute may not be able to diffuse out of this mesh space with a unit step because the random direction of its movement makes it easy to hit the polymer chains. Therefore, it is inappropriate to simply believe that the sieving factor is equal to the percentage of the number of effective meshes to the total number of meshes. As an extreme example, if all of the meshes or areas are uniform and slightly larger than the size or the effective cross-sectional area of the solute, it can be deduced that the sieving factor would be equal to 1 on the basis of previous opinions (not on the basis of the final expression of the model). In fact, we can imagine that when the solute molecules transport in such a network, the polymer chains would be often encountered, which leads to a sieving factor that is absolutely less than 1. 2.2. Model Establishment. To derive a reasonable sieving factor for solute diffusion in hydrogel, taking a single solute molecule in the hydrogel network as an example, the solute repeats Brownian motion, and different-sized meshes would be found in its direction of movement. These mesh sizes have a distribution that can be expressed as the mesh size distribution function of the whole hydrogel network. Meanwhile, there exists a probability that the solute will touch the polymer chain even though the surrounding meshes are larger than the solute size because of the random direction of the solute movement. Considering the above two aspects, we convert the sieving process described before (the solute finds a series of meshes that allow it to pass through) to an intuitive process that a mesh plane sieves a solute (as shown in Figure 1). In detail, there is a mesh plane with a distribution of mesh sizes. The initial position of the solute can be arbitrarily located in a plane that is a unit step away from the mesh plane. It can be derived that the 802


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Figure 3. Schematic process of a solute passing through a layer of mesh or being rebounded. Assuming that the direction of motion of the solute is perpendicular to the mesh plane, only if the distance between the center of the solute and the axis of the polymer chain is greater than the relative distance (a·r*), the solute can pass through the mesh. The values of a for A, B, and C are 1, 1, respectively.

for solute diffusion, as shown in the light blue regions in Figure 2. The total areas of the meshes larger than the solute hydrodynamic size are first calculated and then divided by the total area of all of the meshes (Figure 2A,B, Step 1). The result is the preliminary sieving factor, P1, which can be expressed in eq 2

where b =

Equation 5 (abbreviated as RMP-1 model) represents a rigorous obstruction diffusion model based on the random movement of solute and the intuitive sieving process of mesh plane to the solute. This model is only related to rf, rs, and ξ, and it is noted that the RMP-1 model implicitly ignores the interactions between the solute and the hydrogel network. 2.3. Model Modification. Nevertheless, the physical properties of the solute and the polymer chains and the interaction between them will influence the diffusion behavior of the solute in the hydrogel,16,19,21,41−44 and these should be fully considered during the establishment of the model. In the obstruction model, it is generally believed that the polymer chain is stationary; however, a slight disturbance will be generated in the actual diffusion process, even for the rigid polymer chain. This disturbance comes from the self-diffusion behavior of the polymer chain and the collision of the solute and the polymer chain. It can be hypothesized that this disturbance may have a reaction on the solute diffusion. In Section 2.2, although we assumed that the solute is spherical, it is not completely rigid. Thus, the relationship between the diffusion behavior with the deformation ability of the solute and the polymer chain based on their flexibility should be considered. In addition, the presence of interactions between the solute and the polymer chain, such as the electrostatic, hydrogel bonding, van der Waals force, and hydrophobic interactions, would affect the diffusion of the solute. Therefore, the charge characteristics, hydrophilic property, and other properties of the solute molecule and the characteristic of the polymer chain should also be considered. Besides, the hydration state of the polymer chain may also affect the diffusion behaviors of the solute. In addition to changing the equivalent radius of the polymer chain, the different thickness and strength of the hydrated layer also modulate the reaction force on the polymer chain. Therefore, a variable, a, is introduced into the RMP-1 model, which can reflect the interactions between the solute and the polymer chain. It is assumed that when the distance between the axis of the solute movement direction and the polymer chain central axis is greater than a relative value, a·r*, and the initial direction of movement is perpendicular to the mesh plane (for a fixed diffusion system, a should be a constant and related to the properties of the solute and the polymer chain), the solute can pass through these meshes (Figure 3). Of course, the premise is that the mesh is larger than the solute. It can be assumed that there are two kinds of critical situations in which a solute passes through a mesh. When a < 1,

∞

∫r r 2g (r ) dr D P1 = = s∞ 2 D0 ∫0 r g (r ) dr

(2)

where the expression of P1 is similar regardless of whether the mesh is in the shape of a regular square or a circle. The notation g(r) is the distribution function describing the distribution of the radii of mesh sizes (r) between polymer chains (eq 3), and ξ is the average mesh size40 ⎡ ⎛ ⎞2 ⎤ 2πr r g (r ) = 2 exp⎢ −π ⎜ ⎟ ⎥ ⎝ ⎠ ⎥⎦ ⎢ ξ ξ ⎣

(3)

However, not all of the mesh areas that are larger than the solute hydrodynamic size can be used for solute diffusion because of the possessed obstruction effect of the polymer chains. Previous studies believed that the solute will be rebounded when it encounters a polymer chain. When the solute is passing through a certain mesh, if the distance between the axis of solute movement direction and the central axis of the polymer chain is not less than an apparent critical distance, r* (taken as the sum of the solute radius, rs, and the polymer chain radius, rf), the solute can successfully pass through the mesh. Hence, if the center of the solute is projected onto the yellow area, it would be rebounded. Thus, the effective diffusion area of the solute is the area in light blue (Figure 2C). The second step is to calculate P2, which is the proportion of the effective areas of meshes to the total area of meshes larger than the solute hydrodynamic size (Figure 2B,C, Step 2). And the final form of the sieving factor, P, can be expressed in eq 4 ∞

P = P1·P2 =

∫r r 2g (r ) dr ∫ ∞ (r − (rf + rs))2 g (r ) dr r* s ∞

∫0 r 2g (r ) dr

·

∞

∫r r 2g (r ) dr s

∞

=

∫r * (r − r*)2 g (r ) dr ∞

∫0 r 2g (r ) dr

(4)

Substituting eq 3 into eq 4 2

P = e −b −

π b· erfc(b)

π ·r * . ξ

(5) 803


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The Journal of Physical Chemistry B Scheme 1. Synthesis of PSBMA Hydrogel

r* in the integral function of eq 4 can be directly replaced by a· r*, and a new sieving factor expression is obtained as eq 6 ∞

P = P1·P2 =

yield a 33.3% (w/v) SBMA solution. The initiators (APS and SBS) were added to the SBMA solution with molar percentages of 1.2 and 0.6 mol %, respectively. Then, different amounts of cross-linker (MBAA) were added to the mixture; the molar ratios of SBMA/MBAA were 13, 39, 65, 91, and 117 to prepare the PSBMA hydrogel with different cross-linking densities. All hydrogels were obtained by the polymerization of the precursors at 70 °C for 40 min. 3.3. Equilibrium Swelling of PSBMA Hydrogels. After cross-linking and before swelling, the hydrogels at the relaxation state were weighted (Wr). Then, the hydrogels were swollen in a NaCl solution with different ionic strengths (0, 10, 150, and 300 mM in PBS (10 mM, pH = 7.4)) for 3 days, and the swelling medium was refreshed four times every day. The equilibrium swollen hydrogels were weighted (Ws) and their dry weights were recorded (Wd). Mass swelling ratios of the relaxation (Sr) and equilibrium swelling (Ss) hydrogels were separately calculated by Wr/Wd and Ws/Wd. Volume swelling ratios (Φ) of the hydrogels were expressed as in eq 10 ρpolymer Φ=1+ (S − 1) ρsolvent (10)

∞

∫r * r 2g (r ) dr ∫r * (r − (a·(rf + rs)))2 g (r ) dr ∞

∫0 r 2g (r ) dr

·

∞

∫r * r 2g (r ) dr

∞

=

∫r * (r − a·r*)2 g (r ) dr ∞

∫0 r 2g (r ) dr

(6)

Substituting eq 3 into eq 6 2

2

P = e−b + (1 − a)2 ·b2 ·e−b −

π ab· erfc(b)

(7)

When a > 1, r* in eq 4 can be directly replaced by a·r*, regardless of the integral function or the lower limit of the integral, and the sieving factor can be written as eq 8 ∞

P = P1·P2 =

∞

∫r * r 2g (r ) dr ∫a·r * (r − (a·(rf + rs)))2 g (r ) dr ∞

∫0 r 2g (r ) dr

·

∞

∫r * r 2g (r ) dr

∞

=

∫a·r * (r − a·r*)2 g (r ) dr ∞

∫0 r 2g (r ) dr

where ρpolymer is the polymer density (1.395 g/cm3), ρsolvent is the density of swelling medium, which is approximately equal to 1 g/cm3,47 and S denotes Sr or Ss. The polymer volume fraction of hydrogels at the relaxation (v2,r) and equilibrium (v2,s) states is equal to the reciprocal of the corresponding volume swelling ratio. All experiments were carried out in triplicate, and the results were expressed as means ± standard deviations. 3.4. Compression Tests of PSBMA Hydrogels. After equilibrium swelling in 150 mM swelling medium, the diameter and height of the cylindrical hydrogel were measured. Then, the mechanical strength was measured in an uniaxial compression mode with a crosshead speed of 10 mm/min (WDW-05 electromechanical tester; Time Group Inc, China). The plates were lubricated with silicone grease to prevent friction between the hydrogel and the plates during the loading process. Under small strains (here, 0.93 < λ < 0.96), the shear modulus of the hydrogel could be calculated by eq 1148

(8)

Substituting eq 3 into eq 8 2 2

P = e −a b −

π ab· erfc(ab)

(9)

Equations 7 and 9 (recorded as RMP-2 and RMP-3 models, respectively) represent the established obstruction diffusion models with variable a, and the physical properties of the solute and the polymer network are considered. When a = 1, both eqs 7 and 9 are the same as eq 5, so eq 5 can be considered as a special case of eq 7 or 9.

3. MATERIALS AND METHODS 3.1. Materials. SBMA, sodium metabisulfite (SBS), and ammonium persulfate (APS) were purchased from SigmaAldrich. Phosphate-buffered saline (PBS), sodium chloride, dimethyl sulfoxide (DMSO), N,N′-methylenebisacrylamide (MBAA, 99% purity), and methanol were obtained from Jiangtian Chemical (Tianjin, China). Fluorescein isothiocyanate (FITC), LYZ, FITC−BSA, and FITC−IgG were purchased from Solarbio Biotechnology (Beijing, China). Bipyridine, ethyl-2-bromoisobutyrate (EIBB), and CuBr were provided by Heowins (Tianjin, China). FITC−LYZ was labeled according to the protocol of previous report.45 3.2. Preparation of PSBMA Hydrogels. PSBMA hydrogels were synthesized by free-radical polymerization with redoxinitiation initiators isothermally (Scheme 1).46 In detail, the SBMA hydrogels were dissolved in PBS (pH = 7.4, 10 mM) to

τ=

F = G(λ − λ−2) A

(11)

where τ is the compression stress, F is the compression load, A represents the cross-sectional area of the hydrogel, G denotes the small strain shear modulus, and λ is the compression strain. 3.5. Synthesis of PSBMA Polymer by Atom-Transfer Radical Polymerization. The synthesis of PSBMA was described as follows.49 SBMA (1.5 g, 5.37 mM), EIBB (3.5 mg, 0.018 mM), CuBr (2.55 mg, 0.018 mM), 1,2-bipyridine 804


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Figure 4. Physical properties and parameters of PSBMA hydrogel with different cross-linking densities (or polymer volume fractions). (A) Polymer volume fraction (v2,r and v2,s), (B) shear modulus G, (C) Flory−Huggins parameter χ, and (D) linear relationship between Flory−Huggins parameter and polymer volume fraction. The ionic strength of the swelling medium was 150 mM.

polymer solution concentration based on the Huggins and Kraemer equations. The average value of the reduced viscosity and the inherent viscosity extrapolated to a polymer concentration of zero is taken as the intrinsic viscosity.50 3.7. Gel Permeation Chromatography (GPC). GPC measurements were conducted with a water-based GPC (Viscotek TDA 305; Malvern Instruments) equipped with a triple detector array. PSBMA solutions were prepared in the mobile phase (NaCl solution, 0.1 M, pH = 7.4) at a concentration of 3.83 mg/mL, and the flow rate of the mobile phase was set to 0.5 mL/min. The instrument constants were calibrated by monodisperse poly(ethylene glycol) (99k) standards. All measurements were performed at 30 °C.51 3.8. FRAP. To investigate the diffusion coefficients of proteins in PSBMA hydrogels, the hydrogels were prepared and loaded with different kinds of FITC-labeled proteins (BSA, IgG, LYZ).13 Briefly, 10 μL of hydrogel precursors was added in a square frame fixed on the slide glass. After polymerization, the hydrogel was immersed in 0.2 mg/mL of protein solutions (in 10, 150, and 300 mM NaCl solutions) for 2 days at 25 °C to reach the equilibrium state. Finally, the hydrogels were rinsed with PBS (10 mM, pH = 7.4) and sealed with a cover glass for FRAP measurement, forming a sandwich structure with a hydrogel thickness of less than 100 μm.

(5.62 mg, 0.036 mM), and DMSO (80 mL) were added to a dry Schlenk tube. After three freeze−pump−thaw cycles to remove oxygen, the tube was sealed and placed in preheated oil bath at 30 °C and stirred for 24 h. The polymerization was then quenched by immersing the tube in liquid nitrogen. The PSBMA300 product was precipitated in cold methanol for three times. Then, the collected sample was dried under vacuum for 24 h at 60 °C. According to this method, a series of PSBMA with different molecular weights were synthesized by setting the molar ratio of SBMA/EIBB to 300, 450, 600, and 750. 3.6. Intrinsic Viscosity Measurements. The PSBMA solutions with different concentrations from 0.66 to 0.33 g/dL in 10, 150, and 300 mM NaCl solutions (pH = 7.4) were obtained for all of the polymers with different molecular weights. The intrinsic viscosity of the PSBMA solutions was measured using a capillary Ubbelohde viscometer at 25 ± 0.2 °C. The specific viscosity (ηsp) was calculated from eq 12

ηsp =

t −1 t0

(12)

where t0 is the efflux time of the solvent and t is that of the PSBMA solutions. The value of ηsp is then determined by plotting the reduced viscosity at various polymer solution concentrations and followed by a linear extrapolation to zero 805


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The Journal of Physical Chemistry B Table 1. M̅ c of PSBMA Hydrogels with Different Cross-Linking Densities SBMA/MBAA (mol/mol)

13

39

65

91

117

M̅ c (g/mol)

1169 ± 181

6911 ± 503

11 232 ± 302

15 701 ± 522

19 973 ± 216

To get the predictions of the RMP diffusion models, some parameters are necessary to be figured out, including the chain radius of the PSBMA polymer, the hydrodynamic radius of the protein, and the mesh size of the PSBMA hydrogel. The first two can be obtained from previous studies, but the mesh size is usually obtained via swelling experiments of the hydrogel. The Flory characteristic ratio and Flory−Huggins parameters of PSBMA with swelling media were also calculated to get the mesh sizes of the PSBMA hydrogels. 4.1. Calculation of Flory−Huggins Parameter χ and the Number-Average Molecular Weight of Polymer Chain between Cross-Links (M̅ c). According to the volume swelling ratio of the PSBMA hydrogel, the polymer volume fraction of the hydrogel at relaxation state (v2,r) and equilibrium swelling state (v2,s) were obtained (Figure 4A). The numberaverage molecular weight of the polymer chain between crosslinks (M̅ c) can be calculated by the Peppas−Merrill equation as shown in eq 13.19 Here, M̅ n is the number-average molecular weight in the absence of any cross-linking (because of its large value, the first item on the right-hand side of eq 13 was taken as 0), V1 is the molar volume of the solvent (18 cm3/mol), v ̅ is the specific volume of PSBMA (0.717 cm3/g), and χ is the Flory− Huggins parameter of PSBMA with the swelling medium

The diffusion coefficients of FITC proteins in PSBMA hydrogels were determined by FRAP measurements on a Olympus IX81 microscope coupled with Laser Scanning Confocal Imaging Analysis System (Ultra View Vox; Perkin Elmer) at 25 °C. A 40 mW tunable argon-ion laser operated at 488 nm was used to excite the FITC. The samples were observed using a 10× dry objective lens with a quoted numerical aperture of 0.4. Such a low-numerical-aperture objective lens was chosen to guarantee a cylindrical bleaching volume, enabling diffusion in the third dimension to be avoided. Before bleaching, 10 prebleach images were obtained at 3% maximum laser intensity. Then, a uniform disk with a diameter of 50 μm in the center of the image was bleached at the full laser intensity, and the pulse of bleaching usually took 0.36−0.79 s, which should be sufficiently short to avoid fluorescence recovery during bleaching. After the region of interest (ROI) was brought into focus, a time series of digital images with a resolution of 512 × 512 pixels was recorded using a laser with the same intensity as before. The time intervals between two consecutive images were different. During the first 10 s after bleaching, the interval was set to 50 ms. In the subsequent recovery process, the interval was set to 250 ms. The total recovery time varied from 60 to 120 s on the basis of the protein diffusion rate. Considering the slight bleaching effect of the laser beam on the outside of ROI, it was necessary to correct photobleaching when the fluorescence in ROI was completely recovered, and this correcting process can be achieved by using Velocity 3.0 software. Finally, the diffusion coefficient of protein was directly extracted by the FRAP dataprocessing module. It was worth mentioning that the FRAP model proposed by Soumpasis was chosen to fit the fluorescence recovery curve.52 All FRAP experiments were carried out at 25 °C. To test the reliability of the FRAP measurements and compare with protein diffusions in the PSBMA hydrogels, the diffusion coefficients of protein in 10, 150, and 300 mM swelling media and water were also measured.

2 ] ν ̅ /V1[ln(1 − ν2,s) + ν2,s + χν2,s 1 2 = − 1/3 ⎡ ⎤ M̅ c M̅ n ν ν ν2,r ⎢ ν2,s − 2ν2,s ⎥ 2,r 2,r ⎦ ⎣

( )

(13)

In the PSBMA hydrogel network, the positive and negative charge groups in the polymer chains are quaternary ammonium groups and sulfonic acid groups, respectively. The absolute values of charge density of these two groups are very close.32 Here, we assume that the two kinds of groups have the same ability to absorb the counterions. Therefore, when calculating the Flory−Huggins parameter (χ), the PSBMA hydrogel can be treated as a nonionic hydrogel. After derivation, it can be obtained by eq 14,48 where R is the universal gas constant and T is the absolute temperature (298 K)

4. RESULTS AND DISCUSSION The diffusion of protein or bioactive molecules in zwitterionic hydrogels is significant for its biomedical application. However, as far as we know, few systematic studies focused on the diffusion behaviors of protein in zwitterionic hydrogels. Herein, the diffusion behaviors and diffusion coefficients of proteins (IgG, BSA, and LYZ) in PSBMA hydrogels were investigated. Meanwhile, to validate the RMP models (eqs 5, 7, and 9) on the basis of obstruction theory, PSBMA hydrogels with different mesh sizes obtained by adjusting the cross-linking densities were constructed as models of diffusion matrixes. The finer adjustment of the mesh size of PSBMA hydrogels was realized by controlling the ionic strength of the swelling medium. It is also noted that we considered the physical properties of the solute and the polymer network and the interaction between them in our RMP-2 and RMP-3 models (eqs 7 and 9). This was verified through the establishment of experimental platforms on the basis of the different flexibilities of proteins and the interaction between the proteins and PSBMA chains.

χ=

GV1/(RT ) − ln(1 − v2,s) − v2,s 2 v2,s

(14)

The shear modulus (G) of the PSBMA hydrogels with different cross-linking densities was obtained by stress−strain experiment (Figure 4B). Combined with the polymer volume fractions (Figure 4A) of the PSBMA hydrogels, their Flory− Huggins parameters in the 150 mM NaCl swelling medium are calculated according to eq 14 (Figure 4C). The results show that the interaction parameters between the polymer hydrogel network and the swelling medium are linearly related to the polymer volume fraction under high swelling degree. Figure 4D confirms this for the present system and extrapolates the value of 0.48 to zero polymer volume fraction, which corresponds to the uncross-linked linear polymer at infinite dilution. This is basically consistent with Huglin’s report on KCNS solution.53 Because the M̅ c of PSBMA hydrogels with different crosslinking densities is independent of the subsequent swelling conditions (ionic strength of the swelling medium), it can be 806


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Table 2. Values of Kθ and Cn of PSBMA in NaCl Solutions

calculated only by the swelling data and interaction parameter in 150 mM NaCl solution. As expected, the M̅ c gradually increases with the decrease in the cross-linking density (Table 1). 4.2. Calculation of Flory Characteristic Ratio. The Flory characteristic ratio of PSBMA was calculated by the method of Stockmayer−Fixman equation (eqs 15 and 16)54 [η] M w1/2

= K θ + 0.51ϕ0BM w1/2

K θ = ϕ0(h02 /M w )3/2

(15) (16)

h02 nl 2

(17)

where n is the number of backbone chain bonds and l is the backbone bond length (C−C: 0.154 nm. Combined with eq 16, the Flory characteristic ratio (Cn) can be expressed in eq 18, where Mr is the molecular weight of the repeat unit of PSBMA (279.35 g/mol) ⎛ K ⎞2/3 M Cn = ⎜⎜ θ ⎟⎟ · 2r 2l ⎝ ϕ0 ⎠

10

150

300

Kθ Cn

0.01674 10.91

0.02832 15.49

0.03078 16.37

strength of the swelling media, suggesting that the rigidity or stretching degree of the PSBMA polymer chains increases with the increase in the ionic strength of the swelling media. This performance is consistent with the antipolyelectrolyte effect of zwitterionic polymers. In addition, the Cn value of PSBMA in 150 mM NaCl solution (15.49) is close to that in 300 mM NaCl solution (16.37), which was consistent with the swelling behaviors of PSBMA hydrogels in NaCl solutions with different ionic strengths. 4.3. Modulation of PSBMA Hydrogel Mesh Size and Polymer Chain Radius. The mesh size is a main factor to control the diffusion behavior of the solute in the hydrogel.14 Because the polymer chain of PSBMA has both positive and negative charges, the stretching state of the polymer chain is strongly dependent on the ionic strength of the swelling medium. We prepared PSBMA hydrogels with different mesh sizes by adjusting the cross-linking densities. To realize the fine adjustment of mesh size, the stretching extent of the polymer chain was regulated by changing the ionic strength of the swelling medium. An accurate model matrix of the hydrogel was established for investigating the relationship between the solute diffusion behavior and the hydrogel mesh size. On the basis of the obtained parameters (v2,s, Cn, and M̅ c), the average mesh sizes (ξ) of the PSBMA hydrogel, namely, the end-to-end distance of the polymer chain between cross-links, are calculated by eq 1919 and shown in Table 3

where [η] is the intrinsic viscosity, Mw is the polymer molecular weight of the PSBMA obtained from GPC, φ0 is the Flory viscosity coefficient with a value of 2.1 × 1023 when [η] is expressed in milliliters per gram,55 and h02 is the unperturbed mean-square end-to-end distance. The characteristic ratio (Cn) reflects the stretching degree of polymer chain in the solution. On the basis of polymer solution theory, it can be calculated according to eq 17 Cn =

CNaCl (mM)

(18)

ξ=

Intrinsic viscosities of PSBMA with different molecular weights were obtained by extrapolating the plots of reduced viscosity to zero polymer concentration. Kθ is extracted from Figure 5 after linear fitting of eq 15 and listed in Table 2. Then, the Flory characteristic ratios (Cn) of the PSBMA hydrogel in NaCl solutions (10, 150, 300 mM) are derived from eq 18. As shown in Table 2, the value of Cn increases with increasing the ionic

⎞1/2 ⎛ −1/3 1/2 2M̅ c ν2,s Cn ⎜ ⎟ ·l ⎝ Mr ⎠

(19)

Table 3. Mesh Size (ξ) of PSBMA Hydrogel Calculated via Equation 19 ξ (nm)

a

a

SBMA/MBAA (mol/mol)

10 mM

150 mMa

300 mMa

13 39 65 91 117

2.26 5.66 7.18 8.60 10.06

3.25 9.68 13.59 16.78 19.85

3.51 10.59 15.72 19.44 22.90

Ionic strength of NaCl solution.

Our data suggest that the average mesh sizes (ξ) of the PSBMA hydrogel can be successfully modulated by controlling the cross-linking density and ionic strength of the swelling medium. The ξ of the PSBMA hydrogels increases with the decrease in the cross-linking density in the same swelling medium. Moreover, the ξ also depends on the ion strength of the swelling medium. The higher the ionic strength, the larger the mesh (Table 3 and Figure 6). In the NaCl solution, the electrostatic association of oppositely charged groups of zwitterionic polymer chains would be broken because of the antipolyelectrolyte effect. The degree of disassociation increases with the increase in ionic strength until the complete disassociation, resulting in the increase of ξ with the ionic strength of the swelling medium although the amplitude of

Figure 5. Linear dependence of [η]/Mw1/2 (mL·mol1/2/g3/2) on Mw1/2 of PSBMA in NaCl solutions with different ionic strengths (■: 10 mM; ●: 150 mM; ▲: 300 mM); the intercept (Kθ) of linear fitting results was used in eq 15. 807


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Figure 6. Schematic illustration of the structure of PSBMA hydrogel and the effect of the ionic strength of swelling medium (NaCl solution) on the mesh size (ξ) and equivalent polymer chain radius. When the ionic strength of the swelling medium increases, the mesh size (ξ) of the PSBMA hydrogel increases but the equivalent polymer chain radius (rf) decreases.

variation gradually reduced. Therefore, the adjustable range of the ξ is decided by the cross-linking density. In this range, the ξ can be finely adjusted by changing the ionic strength of the swelling medium. The polymer chain radius (equivalent polymer chain radius) of PSBMA, rf, is obtained by eq 20,15 where NA is the Avogadro number (6.02 × 1023), rf0 is the radius of bare polymer chain, and rw is the radius of a hydration sheath (taken as 0.38 nm34). The value of rf obtained in this study from eq 20 is 0.966 nm ⎛ M rν ⎞1/2 rf = rf0 + rw = ⎜ ⎟ + rw ⎝ 2·lπNA ⎠

Table 4. Properties of LYZ, BSA, and IgG protein LYZ BSA IgG a

Mw (kDa) 14.3 66.4 150.0

pIa 11.2 4.8 7.0

rs (nm)15 1.91 3.63 5.35

flexibility 57

rigid flexible57 flexible56

D0 (μm2/s) 102 ± 458 59.7 ± 4.413 39.3 ± 0.734,59

pI represents the isoelectric point.

sizes (hydrodynamic radius), charged conditions, and flexibilities, which provides the possibility to investigate the relationship between diffusion coefficients of the solute in a hydrogel. First, the diffusion coefficients of proteins of protein in water and NaCl solutions (D0) were determined to examine the reliability of the FRAP experiment. As shown in Figure 7, the diffusion coefficients of all proteins in water are consistent with the values of previous report (cf. Table 4), indicating that the FRAP measurement is reliable and convincing. Moreover, the diffusion coefficients of all of the proteins have no significant difference between water and NaCl solutions with different

(20)

In this diffusion system, we assume that the PSBMA hydrogel network is constructed from long straight fibers. That is to say, the radius of the polymer chain should be the radius of the long straight fiber. In fact, the rf or rf0 in eq 20 is not completely equivalent to the radius of the long straight fiber in our diffusion system. The true equivalence can be achieved only when the PSBMA polymer chains are fully extended. Therefore, eq 20 can be used to calculate the radius of the long straight fiber when the polymer chains fully extend after the hydrogel achieves a swelling equilibrium. However, it is inappropriate in our system because of the following two reasons: (1) not all of the polymer chains can fully extend even if the hydrogel reaches the maximum swelling equilibrium due to the inherent limitation of the polymer network; (2) for the same polymer network, different natures of swelling media (e.g., pH, ionic strength, and temperature) can result in different degrees of swelling equilibrium, further resulting in the inability of the polymer chains to be fully extended. For the diffusion matrix (PSBMA hydrogel), the mesh size and equivalent radius of the polymer chain can be controlled by changing the ionic strength of the swelling medium (Figure 6). However, it is hard to realize the accurate and direct correction radius of the polymer chain. Herein, rf is temporarily taken as 0.966 nm. 4.4. Diffusion Coefficient of Protein Obtained by FRAP. The PSBMA hydrogels with certain ξ (Table 3) are used as the diffusion matrix; IgG, BSA, and LYZ are chosen as the model protein in FRAP measurements. Table 4 displays the related parameters of the model protein. They have different

Figure 7. Diffusion coefficients of IgG, BAS, and LYZ in water and NaCl solutions with different ionic strengths (by FRAP experiment). 808


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Figure 8. FRAP experiment of FITC−BSA diffusion in PSBMA hydrogel with SBMA/MBAA = 117 (mol/mol) and 150 mM ionic strength of the swelling medium. (A) Image stack extracted from the process of fluorescence recovery at different times. (B) The fitting result of normalized fluorescence intensity after corrected bleaching based on the Soumpasis FRAP model.52

values of protein in PSBMS hydrogel are much lower than D0 (Figures 7 and 9) because the protein diffusion may be hindered by polymer chains. In addition, the D values depend on the ξ of the hydrogel and the protein size. For the same protein, D increases with the decrease in the cross-linking density of the hydrogel. IgG, BAS, and LYZ show different D values in the same PSBMA hydrogel. According to the common theory of solute diffusion, if there is no or slightly specific or nonspecific interaction between the diffusion solute and the hydrogel, smaller solute particles would diffuse faster.11 Thus, the D value of proteins in the same PSBMA hydrogel are in the order of LYZ > BSA > IgG. However, under the highest crosslinking density of the PSBMA hydrogel (SBMA/MBAA: 13/1), there is no remarkable difference among the D values of different proteins. In this case, the average ξ of the PSBMA hydrogel is close to or smaller than the protein size, so the protein diffusion is very slowly or even completely blocked. 4.5. Predictive Ability of RMP-1 Model on the Protein Diffusion in PSBMA Hydrogels. Taking the radius of polymer chain, rf, as 0.966 nm and rs as shown in Table 4, the normalized diffusivities, D/D0, of BSA, LYZ, and IgG in PSBMA hydrogels are predicted by the RMP-1 model (eq 5) and the Amsden model (eq 1B), respectively. As shown in Figure 10, both these models can reflect the diffusion trends of different proteins in PSBMA hydrogels with different mesh sizes; however, our established RMP-1 model can more accurately predict the D/D0 than the Amsden model. The obstruction model developed by Amsden is according to the idea that solute molecules passing through a hydrogel network are governed by the probability of the solute finding a succession of openings between the polymer chains, and the opening is larger than the hydrodynamic size of the solute. The diffusivity or sieving factor equals the percentage of the number of meshes larger than the solute hydrodynamic size to the total number of the meshes. Therefore, according to Amsden’s opinions (not based on the final expression of Amsden model),14 if all of the meshes of the hydrogel network are

ionic strengths. Therefore, the electrostatic interactions between the protein and the ions of the solution medium have little effects on the diffusion behaviors of the protein. Then, the diffusion coefficients of proteins in the PSBMA hydrogels (D) were investigated by FRAP experiment. Because of the obstruction effect of polymer chains, protein diffusion in the PSBMA hydrogel is significantly slower than that in solution. The fluorescence recovery process can be seen clearly in a longer time scale. A typical FRAP experiment of FITC− BSA diffusion in PSBMA hydrogel is illustrated in Figure 8. After the acquisition of the image stacks, D is determined by FRAP analysis of the Soumpasis equation to the fluorescence recovery curve (Figure 8B). Figure 9 shows the D values of IgG, BAS, and LYZ in PSBMA hydrogels with different cross-linking densities. The D

Figure 9. Diffusion coefficients (D) of proteins in NaCl solution and PSBMA hydrogels with different cross-linking densities (by FRAP experiment). The swelling medium of the hydrogels was 150 mM NaCl solution. 809


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Figure 10. Normalized diffusivity (D/D0) vs mesh size for protein diffusion in PSBMA hydrogels with 150 mM swelling medium; the solid lines represent the prediction values and the discrete points represent the experimental values. (A) RMP-1 model predictions comparing with experiments. (B) Amsden model predictions comparing with experiments.

models. Moreover, the ionic strength of the swelling medium can effectively adjust the D value, especially at low cross-linking density, because the ionic strength of the swelling medium can modulate the ξ of the PSBMA hydrogel. The PSBMA hydrogel cannot fully swell in the medium with low ionic strength owing to the electrostatic association interaction between the negative sulfonic acid groups and the positive quaternary ammonium groups in the PSBMA chain. However, the charge-screening effect would increase with the increase in the ionic strength. This performance can induce the increase of swelling degree, lead to the outstretching of PSBMA chains, and increase the ξ of the hydrogel, thereby further improving the diffusion capacity of proteins in the PSBMA hydrogels. According to the fitting results, the variable a is more than 1 for LYZ and less than 1 for BSA and IgG regardless of fitting from the RMP-2 or RMP-3 model. The RMP-2 and RMP-3 models are suitable for a < 1 and > 1, respectively. Thus, the fitting results of the RMP-3 model are used for LYZ and the fitting results of the RMP-2 model are used for BSA and IgG. Meanwhile, the value of the variable a and related fitting parameters are listed in Table 5. Results suggest that the RMP2 and RMP-3 models can well fit the experimental values. Almost all of the values of R2 are more than 0.95. Although the two models are applicably different, the difference of the fitting result is small. Comparing the Amden model with our BMP-1 model, the fitting results of the RMP-2 and RMP-3 models are closer to the experimental values because of the introduction of the variable a. It is assumed that when the distance between the axis of the solute movement direction and the polymer chain central axis is larger than a relative value (a·r*, r* = rs + rf), the solute can pass through the mesh, provided ξ is larger than the solute size. When a is less than 1, that distance is less than r*, and the solute is likely to pass the mesh. On the contrary, when a is more than 1, that distance is larger than r*, and the solute cannot possibly pass through the mesh. Herein, the values of a fitted by the normalized diffusivity of IgG, BSA, and LYZ are recorded as aIgG, aBSA, and aLYZ, respectively. As shown in Table 5, aIgG and aBSA are less than 1, but aLYZ is larger than 1. It is well known that BSA and IgG are flexible proteins, and they could easily deform when they encounter obstacles (polymer chains). Therefore, it is possible for BSA

larger than the hydrodynamic size of the solute, the normalized diffusivity (D/D0) of the solute would equal 1. This is not reasonable because the polymer chains hindered some random movement of the solute. Thus, the Amsden model overestimated the normalized diffusivity (D/D0), and the prediction based on this model is larger than the experimental results, especially with large meshes (Figure 10B). On the contrary, in the RMP-1 model, we fully consider the randomness of the solute movement in the hydrogel network. The sieving process is converted to a more intuitive manner, which is described in the obstruction effect of mesh plane to the solute. It overcomes the problem that the marked obstruction effect of polymer chains is neglected if the surrounding meshes are larger than the solute size in the Amsden model. Hence, the RMP-1 model presents a better predictive ability than the Amsden model. Especially, both the Amsden and RMP-1 models assume that the solute is a hard sphere, polymer chains are immobile, and the interactions between the polymer chains and solute are neglected. Although the RMP-1 model is relatively more accurate, there is still a deviation between the predictions and the experimental values. The predictions of LYZ are slightly higher than the experimental values but lower for IgG and BSA. This may be associated with the solute properties and the interaction between the protein and the PSBMA chains. 4.6. Assessment of RMP-2 and RMP-3 Models and Analysis of Protein Diffusion Behavior in PSBMA Hydrogels. To achieve more accurate predictive ability and to explore a more realistic diffusion situation for solute in hydrogel, the physical properties of the solute and the polymer network and the interaction between them are considered. The variable a is introduced in the RMP-2 (eq 7) and RMP-3 (eq 9) models. In addition, we further evaluated the D value of proteins in PSBMA hydrogels in the swelling media with different ionic strengths, trying to find out the effects of the physical properties (flexibility, charge, etc.) of proteins and hydrogel network on the protein diffusion behavior and recognize the relationship between the value of a and the diffusion behavior of solute in the hydrogel. As shown in Figure 11, the normalized diffusivities (D/D0) of all proteins in the PSBMA hydrogels with swelling media of different ionic strengths are fitted with the RMP-2 and RMP-3 810


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Figure 11. Normalized diffusivity (D/D0) vs mesh size (ξ) for protein diffusion in PSBMA hydrogels with swelling media of different ionic strengths (A, D: 10 mM; B, E: 150 mM; C, F: 300 mM). All experimental data were fitted with RMP-2 (A−C) and RMP-3 (D−F) models. The solid lines represent fitting results of the RMP-2 or RMP-3 model, and the discrete points represent the experimental values.

and IgG to pass through the mesh even when the distance of the axis of protein movement direction to the PSBMA chain central axis is less than r*. In other words, even if BSA and IgG encounter the polymer chain, they might pass through the mesh. Therefore, aIgG and aBSA are less than 1. LYZ has no obvious deformability because of its rigid nature when it encounters obstacles. That means, aLYZ should not be less than

1. As for the interaction between the solute and the polymer network, the electrostatic interaction between the protein and the PSBMA chains may play a key role. The end of the side group of the PSBMA chain is negatively charged. Hence, the solute with a positive charge would be subject to electrostatic attraction when getting close to the PSBMA chain. That leads to an increase in the drag force between the PSBMA chain and 811


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interaction between the protein and the PSBMA hydrogel network on the diffusion behavior of the protein. It is necessary to note that the universality of our model should be further validated by more diffusion systems, and the physical meaning of the variable a also needs to be further studied in detail. However, this model provided a preliminary guidance for the study of diffusivity and permeability adjustment of proteins in zwitterionic hydrogels, and it will be conducive to the practical application of the zwitterionic hydrogels in the process of biological transport.

Table 5. Model Variable (a) Obtained through Fitting the Experimental D/D0 with RMP-2 and RMP-3 Models protein IgGa

BSAa

LYZb

a

CNaCl (mM)

model variable (a)

S.E.

coefficient of determination (R2)

10 150 300 10 150 300 10 150 300

0.86686 0.82062 0.80413 0.87168 0.84407 0.81909 1.12027 1.05416 1.04201

0.04408 0.01644 0.03843 0.01021 0.05371 0.01539 0.03409 0.03371 0.04108

0.69245 0.98958 0.94089 0.99253 0.93005 0.99432 0.95441 0.98728 0.98397

■

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected], [email protected]. Tel: +8610-68166874 (J.L.). *E-mail: [email protected]. Tel: +86-22-27402893 (F.Y.).

Fitting with RMP-2 model. bFitting with RMP-3 model.

the positively charged solute molecule and makes the solute difficult or fail to pass through the mesh. As a result, when the positively charged protein passes through the mesh, the distance of the axis of protein movement direction to the polymer chain central axis should be larger than r* to get rid of the electrostatic force, that is, a > 1. LYZ is positively charged in a medium with pH of 7.4, resulting in aLYZ > 1. In addition, a value decreases with the increase in the ionic strength of the swelling medium. In the PSBMA hydrogel, when the ionic strength of the medium increases, charge screening is more obvious, the state of the polymer chain will gradually change from shrinkage to stretching, and the equivalent radius of the polymer chain (rf) will significantly reduce (Figure 4). On the basis of apparent analysis, the solute can more easily pass through the mesh, leading to the decrease in a value with the increase in the ionic strength. However, this change is not significant because rf is much less than the mesh size (ξ) or the radius of the protein (rs). Notably, the PSBMA chain is relatively rigid according to the Flory characteristic ratio, which is consistent with our hypothesis. Moreover, the obstruction model is suitable for the situation of the solute diffusion in a relatively rigid hydrogel network,34 so the obstruction model is applicable for the prediction of solute diffusion in PSBMA hydrogels, especially when the polymer chains are extended. It may be one of the reasons that our model can predict the diffusion coefficient accurately.

ORCID

Fanglian Yao: 0000-0003-4153-3543 Author Contributions

The article was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Funding

This work is supported by the National Nature Science Foundation of China (51573127, 31271016, and 31370975) and Beijing Natural Science Foundation (No. 7162150). Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors thank Prof. Wenguang Liu at Tianjin University for performing mechanical strength test and Nannan Xiao at Nankai University for providing the guidance of FRAP measurements.

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REFERENCES

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5. CONCLUSIONS A series of PSBMA hydrogels with different mesh sizes were prepared by changing the feed ratios of SBMA and MBAA. The mesh size can be finely adjusted by changing the ionic strength of the swelling medium. The diffusion behavior of protein in the PSBMA hydrogel strongly depended on the mesh size and the ionic strength of the swelling medium. Considering the randomness of solute movement based on obstruction theory, we established the RMP-1 model. Furthermore, taking into account the physical properties of the solute and the hydrogel network and the interaction between them, the RMP-2 and RMP-3 models were proposed. It showed that the RMP-1 model has good predictive ability for IgG, BSA, and LYZ diffusion in PSBMA hydrogels compared with the FRAP experimental results. In addition, the predictions of the RMP-2 and RMP-3 models are closer to the experimental values, different from that of the RMP-1 model, due to the introduction of the variable a. And the values of a fitting with the RMP-2 or RMP-3 model can reveal the effect of the 812


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