Establishment of a Physical Model for Solute Diffusion in Hydrogel

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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 J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.6b10355 • Publication Date (Web): 06 Jan 2017 Downloaded from http://pubs.acs.org on January 7, 2017

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The Journal of Physical Chemistry

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,§ Fanglian Yao†,⊥, * †

School of Chemical Engineering and Technology, 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 ⊥

Key Laboratory of Systems Bioengineering, Ministry of Education, Tianjin University, Tianjin

300072, China

ACS Paragon Plus Environment

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ABSTRACT: The 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 hydrogel network, a physical diffusion RMP-1 model based on obstruction theory was established in this study. Based on this, the physical properties of the solute and polymer chain and the interactions between them were introduced into this model, RMP-2 and RMP-3 model 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. Flory characteristic ratios, Flory-Huggins parameter, mesh size and polymer chain radii of PSBMA hydrogels were determined. The diffusion coefficients of the proteins (albumin from bovine serum (BSA), Immunoglobulin G (IgG) and Lysozyme) in PSBMA hydrogels were studied by fluorescence recovery after photobleaching (FRAP) 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 PSBMA network. It was also confirmed that the diffusion behaviors of protein in zwitterionic hydrogel can be adjusted by changing the crosslinking density of the hydrogel and the ionic strength of the swelling medium. Our model will be expected to possess accurately predictive ability for the diffusion coefficient of solute in hydrogel, which will be widely used in the biomedical field.


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1. INTRODUCTION Hydrogels are often regarded as biomaterials because of their ability to contain large amounts of water and to display characteristics similar to natural tissues.1,

2

Furthermore, their three-

dimensional (3-D) network structure is extremely suitable for accommodating bioactive compounds, such as therapeutically proteins and peptides,3 and even live cells,4 which renders them used for protein delivery, protein separation, therapeutic implant and cell encapsulation.5-7 Almost all of these applications involve protein diffusion, and the diffusion behavior of protein in hydrogel will directly affect the applications efficiency. Taking cell encapsulation as an example, in order 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 attentions. The researchers not only focused on the experimental investigation of the diffusion process and characteristics, but also established a series of physical models to predict the diffusion coefficient.10-21 Recently, zwitterionic hydrogels have been widely used in biomedical applications due to their high hydration and ultra-low fouling properties,22,23 such as drug delivery carriers,24,25 in vivo implants and cell encapsulation matrixes, etc.26-30 Poly (carboxybetaine methacrylate) (PCBMA) hydrogel after modified with additional cell-adhesion moieties can provide an excellent 3-D environment for cell growth, which was a potential carrier for cell encapsulation.2630

Ishihara et al. found that a cytocompatible phospholipid polymer hydrogel can be used as a 3-

D cell encapsulation culture matrix, and the proliferation of cells encapsulated in this hydrogel can be modulated by the concentrations of polymers and the swelling degree of hydrogel.27 The further understanding of the diffusion behaviors of proteins or other solutes in these zwitterionic


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hydrogels will provide a clearer guidance for their applications in biomedical field. Different from other hydrophilic polymer hydrogels, zwitterionic hydrogels are characterized by their antipolyelectrolyte behavior. For example, zwitterionic polymer chain presents shrinkage state in water, but stretches in aqueous solution in the presence of salt ions, which provides the possibility to adjust the mesh size of hydrogel and the flexibility of polymer chain by changing the ionic strength of swelling medium. In addition, many of 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 stimuli-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 few systematic studies on protein diffusion in zwitterionic hydrogels, even though the broad applications of the zwitterionic hydrogel in 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, providing guidance for the structural design of hydrogels in biomedical fields. Specifically, zwitterionic PSBMA hydrogels were selected as the model diffusion matrixes. The mesh sizes of PSBMA hydrogels in a wide range were realized by changing the crosslinking density and the fine adjustable mesh sizes were obtained through controlling the ionic strength of swelling medium. The network characteristics and related parameters of 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


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of proteins in PSBMA hydrogels were determined by fluorescence recovery after photobleaching (FRAP) measurements and the results were used for model validation. Furthermore, the comparison between model predictions and experimental values was analyzed, attempting to reveal how the properties of protein and PSBMA network and the interaction between them affect the diffusion behaviors of protein in the PSBMA hydrogel.

2. MODEL DEVELOPMENT There are lots of physical models used for solute diffusion in polymer hydrogel,33-35 and many of them are based on obstruction theory.14-17,

36-39

However, the physical properties of

solute or 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 can’t be reflected in these models. In addition, the naturally random movement of the solute in hydrogel network should be comprehensively considered. Therefore, in this section, we took account of the self-diffusion characteristics of the solute in the hydrogel network and established a new diffusion model based on obstruction theory, attempting to accurately predict the diffusion coefficient of a solute in hydrogel. 2.1 Theory 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 polymer mesh with appropriate size. So the polymer chains play a role of sieving the solute that can only pass the mesh between the adjacent polymer chains.33,34 The establishment of the physical model based


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on obstruction theory was 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 was expressed in Eq. (1A).

 (r + r )  D = exp− s f v2, s  (1A) D0 rf   in which D0 and D are the solute diffusion coefficient in aqueous solution and hydrogel, respectively, rs and rf are the radius of solute and polymer chain, respectively. v2,s is the polymer volume fraction of the hydrogel at 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 their hydrodynamic radii might be different. Sieving factor is a function of the proportion of the hydrodynamic radius of the solute and the average mesh size of the hydrogel.18 Based on Lustig and Peppas's view, Amsden also believed that the sieving factor should be the percentage of the number of meshes whose size was 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, in which P is the sieving factor, and r is the average radius of the openings between the polymer chains.


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 π r +r P= = exp−  s f D0  4  r + rf  D

   

2

   

(1B)

Therefore, obstruction diffusion models can be divided into two categories: models based on the relationship of effective area and 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. And they did not take into account whether the probability this mesh contributed to the sieving factor had deeper relationship with the size or area of the mesh. Because there is a mesh size distribution in the hydrogel, some of the meshes are significantly larger than the solute. It can be imagined that the probability contributed by these meshes should be larger than that of smaller meshes. It can be imagined that even when all the mesh sizes 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. So it is inappropriate to simply believe that the sieving factor is equal to the percentage of the number of effective meshes to total meshes. As an extreme example, if all of the mesh sizes or areas are uniform and slightly larger than the size or effective crosssectional area of the solute, it can be deduced that the sieving factor would be equal to 1 based on previous opinions (not based on 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. In order 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


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movement. These mesh sizes have a distribution which 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 mesh sizes are larger than the solute size, due to the random direction of the solute movement. Considering the two aspects above, 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 sieving factor equals the probability that the solute can pass through the mesh plane from any position in this plane.

Figure 1. Schematic diagrams of convert process of 3D network sieve solute to 2D mesh plane. For the diffusion system and its screening process, there are several assumptions as following. 1) The hydrogel is composed of a long straight polymeric fiber network with random orientation; 2) The distribution function of the mesh size can be expressed by the radius distribution function of spherical space in long straight polymer fiber network with random orientation, which was proposed by Ogston;40 3) The solute is considered to be spherical, and its


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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 different initial positions of the solute. This would fully consider the randomness 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, so 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 polymer chain radius, and focus on the mesh regions that can be used for solute diffusion, as shown in the light blue regions in Figure 2. The total areas of the meshes whose sizes are larger than the solute hydrodynamic size is first calculated and then divide by the total area of all the meshes (Figure 2A&B. Step 1). The result is the preliminary sieving factor P1 and it can be expressed by Eq. (2):

D P1 = = D0

∫ ∫

∞

rs ∞

0

r 2 g ( r ) dr

(2) r 2 g ( r ) dr


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in which, the expression of P1 is similarly regardless whether the mesh is in regular square or circle. g(r) is the distribution function describing the distribution of the radii of mesh sizes (r) between polymer chains (Eq. (3)). ξ is the average mesh size.40

2π r

g (r ) =

ξ2

r ⋅ exp[ −π  ξ

2

  ] (3) 

However, not all the mesh areas whose size is larger than solute hydrodynamic size can be used for solute diffusion, due to 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 solute radius, rs, and polymer chain radius, rf), the solute can successfully pass through the mesh. Hence, if the center of the solute is projected on 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 effective areas of these meshes to all the areas of those meshes whose size is larger than solute hydrodynamic size (Figure 2B&C. Step 2). And the final form of the sieving factor P can be expressed as Eq. (4): ∞

∞

∫ r g(r)dr ⋅ ∫ P= P ⋅P = ∫ r g(r)dr 1

2

rs ∞

2

r

2

0

*

(r − (rf + rs ))2 g (r)dr ∞

∫

rs

∞

∫ =

r 2 g (r)dr

r*

(r − r * ) 2 g (r)dr ∞

∫

0

r 2 g (r)dr

(4)

Substituting Eq.(3) into Eq.(4): 2

P = e−b − π b ⋅ erfc(b)

(5)


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in which, b =

π ⋅ r* . ξ

Eq. (5) (abbreviated as RMP-1 model) represenst a rigorous obstruction diffusion model based on the random movement of solute and the intuitive sieving process of mesh plane to solute. This model is only related to rf, rs and ξ. And it is noted that the RMP-1 model implicitly ignore the interactions between solute and hydrogel network.

Figure 2. Schematic diagrams of the process of model establishment (A, B, C). Step1 indicates that there is a distribution function of the hydrogel mesh size and only a part of the meshes are able to diffuse for a certain size solute. Step 2 shows that only when the centre of solute projects in light blue areas that the solute can pass through the mesh. 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 hydrogel,16,19,21,41-44 and these should be fully considered during the model establishment. In the obstruction model, it is generally believed that the polymer chain is stationary, but 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


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chain and the collision of solute to 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, if there are interactions between the solute and the polymer chain, such as electrostatic, hydrogel bonding, Van der Waals force and hydrophobic interactions etc., it would affect the diffusion of the solute. So the charged characteristics, hydrophilic property and other properties of the solute molecule and the characteristic of 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 of a is introduced into the RMP-1 model, which can reflect the interactions between solute and polymer chain. It is assumed that when the distance between the axis of solute movement direction and the polymer chain central axis is larger than a relative value a·r* and the initial direction of motion 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 polymer chains), the solute can pass through these meshes (Figure 3). Of course, the premise is that the mesh size is larger than the solute size. It can be imagined that there are two kinds of critical situations about solute passing through the mesh. When a< 1, r* in the integral function of the Eq. (4) can be directly replaced by a·r*, and a new sieving factor expression is obtained as Eq. (4).


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∞

P = P1 ⋅ P2 =

∞

2 ∫ * r g(r)dr r ∞

∫

0

2 ∫ * (r − (a ⋅ (rf + rs ))) g(r)dr

⋅

r

∞

2 ∫ * r g(r)dr

r 2 g (r)dr

∞

∫ =

r*

(r − a ⋅ r * ) 2 g (r)dr ∞

∫

0

r

(6)

r 2 g (r)dr

substituting Eq.(3) into Eq.(6):

2

2

P = e −b + (1 − a ) 2 ⋅ b 2 ⋅ e −b − π ab ⋅ erfc (b)

(7)

When a > 1, r* in the Eq. (4) can be directly replaced by a·r*, no matter in the integral function or the lower limit of the integral, and the sieving factor can be written as Eq. (8) ∞

P = P1 ⋅ P2

∫ = ∫

*

r ∞

0

r 2 g (r )dr

∞

∫ ⋅

a⋅r

*

(r − (a ⋅ (rf + rs )))2 g (r )dr ∞

2 ∫ * r g(r)dr

r 2 g (r)dr

∞

∫ =

a⋅r*

(r − a ⋅ r * )2 g(r)dr

r

∞

∫

0

r 2 g (r)dr

(8)

substituting Eq.(3) into Eq.(8): 2 2

P = e − a b − π ab ⋅ efrc(ab)

(9)

Eqs. (7) and (9) (record as RMP-2 and RMP-3 model, 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 of 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).


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

3. MATERIALS AND METHODS 3.1 Materials. Sulfobetaine methacrylate (SBMA), sodium metabisulfite (SBS) and ammonium persulfate (APS) were purchased from Sigma–Aldrich (USA). Phosphate buffered saline (PBS), sodium chloride, dimethyl sulfoxide (DMSO), N, N’-Methylene-bisacrylamide (MBAA, 99% purity) and Methanol were obtained from Jiangtian Chemical (Tianjin, China). Fluorescein isothiocyanate (FITC), Lysozyme (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.

Poly (sulfobetaine methacrylate) (PSBMA)

hydrogels were synthesized by free-radical polymerization with redox-initiation initiators


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isothermally (Scheme 1).46 In detail, the SBMA were dissolved in PBS (pH=7.4, 10 mM) to yield 33.3 % (w/v) SBMA solution. The initiators (APS and SBS) were added in SBMA solution, their molar percentage to SBMA is 1.2 mol % and 0.6 mol %, respectively. Then different amounts of crosslinker (MBAA) were added in the mixture, the molar ratios of SBMA/MBAA were 13, 39, 65, 91 and 117 to prepare PSBMA hydrogel with different crosslinking densities. All hydrogels were obtained by polymerization of the precursors at 70 °C for 40 min.

Scheme 1. Synthesis of PSBMA hydrogel 3.3 Equilibrium Swelling of PSBMA Hydrogels. After crosslinking and before swelling, the hydrogels at relaxation state were weighted (Wr). Then hydrogels were swollen in NaCl solution with different ionic strengths (0 mM, 10 mM, 150 mM and 300 mM in PBS (10 mM, pH=7.4)) for 3 days and the swelling media was refreshed 4 times every day. The equilibrium swollen hydrogels were weighted (Ws) and its dry weight was recorded as Wd. Mass swelling ratios of relaxation (Sr) and equilibrium swelling (Ss) hydrogel were separately calculated by Wr/Wd and Ws/Wd. Volume swelling ratios (Φ) of the hydrogels were expressed as in Eq. (10):

Φ = 1+

ρ polymer ( S − 1) ρ solvent

(10)


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in which, ρpolymer is the polymer density (1.395g/cm3), ρsolvent is the density of swelling medium and is approximately equal to 1 g/cm3.47 S denotes Sr or Ss. The polymer volume fraction of hydrogels at the relaxation (v2,r) and the equilibrium state (v2,s) 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 the height of the cylindrical hydrogel were measured. Then, the mechanical strength was measured by 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. Under small strains (Here, 0.93 IgG. But under the highest crosslinking density of PSBMA hydrogel (SBMA/MBAA: 13/1), there are no remarkable differences among the D of different proteins. In this case, the average ξ of PSBMA hydrogel is close to or smaller than the protein size, so the protein diffusion of protein is very slowly and even being completely blocked.


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Figure 9. Diffusion coefficients (D) of proteins in NaCl solution and PSBMA hydrogels with different crosslinking densities (by FRAP experiment). The swelling medium of hydrogels was 150 mM NaCl solution. 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 diffusivity D/D0 of BSA, LYZ and IgG in PSBMA hydrogels are predicted by RMP-1 model (Eq. (5)) and Amsden Model (Eq. (1B)), respectively. As shown in Figure 10, both of the two models can reflect the diffusion trends of different proteins in PSBMA hydrogels with different mesh sizes, but our established RMP-1 model can more accurately predict the D/D0 than 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 size is larger than the hydrodynamic size of the solute. The diffusivity or sieving factor equals the percentage of the number of meshes whose size is 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 mesh sizes of the hydrogel network are 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 Amsden model is larger than experimental results, especially under large mesh sizes (Figure 10B). On the contrary, in RMP-1 model, we fully consider the randomness of the solute movement in the hydrogel network. The sieving process is converted to a more


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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 solute size in the Amsden model. Hence, the RMP-1 model presents a better predictive ability than Amsden model. Especially, both of the Amsden and RMP-1 models assume that solute is hard sphere, polymer chains are immobile and the interactions between 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 they are lower for IgG and BSA. This may be associated with the solute properties and the interaction between the protein and the PSBMA chains.

Figure 10. The normalized diffusivity D/D0 is plotted versus mesh size for proteins 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.


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4.6 Assessment of

RMP-2 and RMP-3 Models and Analysis of Protein Diffusion

Behavior in PSBMA Hydrogels. In order 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 RMP-2 model (Eq. (7)) and RMP-3 model (Eq. (9)). In addition, we further evaluated the D 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 hydrogel. As shown in Figure 11, the normalized diffusivities (D/D0) of all proteins in PSBMA hydrogels with swelling media of different ionic strengths are fitted with RMP-2 and RMP-3 models. Moreover, the ionic strength of swelling medium can effectively adjust the D, especially at low crosslinking density, because the ionic strength of swelling medium can modulate the ξ of PSBMA hydrogel. PSBMA hydrogel can’t 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 PSBMA chain. However, the charge screening effect would increase with the increase of ionic strength. This performance can induce the increasing of swelling degree, lead to the outstretching of PSBMA chains and increase the ξ of hydrogel, which further improve the diffusion capacity of proteins in PSBMA hydrogels.


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Figure 11. The normalized diffusivity D/D0 is plotted versus mesh size (ξ) for proteins 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 (ABC) and RMP-3 (DEF) model. The solid lines represent fitting results of RMP-2 or RMP-3 model, and the discrete points represent the experimental values. According to the fitting results, the variable a is more than 1 for LYZ and is less than 1 for BSA and IgG, no matter fitting from RMP-2 or RMP-3 model. The RMP-2 and RMP-3 model is suitable for a 1, respectively. Thus, the fitting results of RMP-3 are used for LYZ and the fitting results of RMP-2 are used for BSA and IgG. Meanwhile, the value of the variable a


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and related fitting parameters are listed in Table 5. Results suggest that the RMP-2 model and RMP-3 model can well fit the experimental values. Almost all 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 with Amden model and our BMP-1 model, the fitting results of RMP-2 model and RMP-3 model are closer to the experimental values due to the introducing of 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), solute can pass the mesh with the premise that the ξ is larger than solute size. When a is less than 1, that distance is less than r*, the solute is likely to pass the mesh. On the contrary, when a is more than 1, that distance is larger than r*, it cannot possibly pass through. Herein, the values of a fitting 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 and IgG to pass through the mesh even when the distance of the axis of protein movement direction to PSBMA chain central axis is less than r*. In other words, even if BSA and IgG encounters the polymer chain, they might pass through the mesh. So aIgG and aBSA are less than 1. LYZ has no obvious deformability due to 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 PSBMA chains may play a key role. The end of the side group of the PSBMA chain is negative charged. Hence, the solute with positive charge would be subject to electrostatic attraction when getting close to PSBMA chain. That leads to the increase of drag force between PSBMA chain and the positive charged solute molecule and makes the solute difficult or failed


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to pass through the mesh. As a result, when the positive charged protein passing through the mesh, the distance of the axis of protein movement direction to the polymer chain central axis should be larger than r* in order to get rid of the electrostatic force, i.e. a>1. LYZ is positive charged in the medium with the pH of 7.4, resulting in aLYZ>1. Table 5. The Model Variable (a) Obtained through Fitting the Experimental D/D0 with RMP-2 Model and RMP-3 Model CNaCl Model variable Protein S.E Coefficient of determination (R2) (mM) (a) 10 0.86686 0.04408 0.69245 p IgG 150 0.82062 0.01644 0.98958 300 0.80413 0.03843 0.94089 10 0.87168 0.01021 0.99253 BSAp 150 0.84407 0.05371 0.93005 300 0.81909 0.01539 0.99432 10 1.12027 0.03409 0.95441 LYZq 150 1.05416 0.03371 0.98728 300 1.04201 0.04108 0.98397 p q fitting with RMP-2 model, fitting with RMP-3 model.

In addition, a value decreases with the increasing of ionic strength of swelling medium. In PSBMA hydrogel, when the ionic strength of medium increases, charge screening is more obvious, the state of polymer chain will gradually change from shrinkage to stretching, and the equivalent radius of polymer chain (rf) significantly reduce (Figure 4). Based on apparent analysis, the solute is more easily to pass through the mesh, leading to the decrease of a value with the increase of the ionic strength. But that is not significant because rf is much less than the mesh size (ξ) or radius of 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 relatively rigid hydrogel network,34 so obstruction model is


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applicable for the prediction of the 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.

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 swelling medium. The diffusion behavior of protein in PSBMA hydrogel strongly depended on the mesh size and the ionic strength of swelling medium. Considering the randomness of solute movement based on obstruction theory, we established the RMP-1 model. Furthermore, taking account of the physical properties of solute and hydrogel network and the interaction between them, 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 RMP-2 and RMP-3 model are closer to the experimental values than RMP-1, due to the introduction of the variable a. And the values of a fitting with RMP-2 or RMP-3 model can reveal the effect of the 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. In spite, this model provided a preliminary guidance for the study of the diffusivity and permeability adjustment of proteins in zwitterionic hydrogel, and the model will be conducive to the practical application of the zwitterionic hydrogel in the process of biological transport.


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AUTHOR INFORMATION Corresponding Author *Email:

[email protected],

Tel:

+86-22-27402893

(F.

L.,

Yao);

Email:

[email protected], [email protected], Tel: +86-10-68166874; (J. J, Li) Author Contributions The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Funding Sources This work is supported by National Nature Science Foundation of China (51573127, 31271016 and 31370975) and Beijing Natural Science Foundation (No. 7162150). ACKNOWLEDGMENT Authors would like to thank the 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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Establishment of a Physical Model for Solute Diffusion in Hydrogel

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