Modeling the Mechanical Response of Tetragonal Lysozyme Crystals

Dec 29, 2009 - Probing biomechanical properties with a centrifugal force quartz crystal microbalance. Aaron Webster , Frank Vollmer , Yuki Sato...
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Modeling the Mechanical Response of Tetragonal Lysozyme Crystals Amir Zamiri* and Suvranu De Mechanical, Aerospace, and Nuclear Engineering Department, Rensselaer Polytechnic Institute, 110 Eighth St., Troy, New York 12180 Received September 3, 2009. Revised Manuscript Received December 10, 2009 We investigate the temperature- and humidity-dependent mechanical response of tetragonal lysozyme crystals using a continuum-based crystal plasticity model calibrated with existing experimental data. The effects of temperature and intracrystalline water are incorporated in the elastic constant of the protein crystal as well as the critical resolved shear stress on the slip planes in the crystal plasticity model. The critical resolved shear stresses have been deduced from Vickers microhardness test data corresponding to different temperatures and states of hydration. Compression analyses have then been carried out along different crystallographic directions of lysozyme crystals which reveal that their mechanical response is highly anisotropic and orientation dependent, being purely elastic along the [110] direction but elastoplastic along the [100] and [212] directions. An interesting observation is that an increase in temperature and the amount of intracrystalline water molecules leads to a decrease in the critical resolved shear stress of the slip systems resulting in softening of the crystal. The analysis presented in this paper may be applied to the study of other protein crystal systems as well as their optimal design for biotechnological applications.

1. Introduction Proteins crystals are highly ordered three-dimensional structures, in which the protein molecules bind to each other with specific intermolecular interactions. In the past, the main goal of protein crystallization was to explore the structure of protein molecules using X-ray and electron microscopy.1 Recently, protein crystals have emerged as promising bionanoporous materials for different applications including highly selective biocatalysis, biosensing, bioseparation, vaccine formulation, and drug delivery.1-4 Over the past decade, significant research and development efforts have been focused on engineering protein crystals, efficacy testing, model development, and production and characterization.7-10 Despite these successes, many challenges associated with the characterization of protein crystals including stability remain. The environmental working conditions require the protein crystals to be both chemically and mechanically stable. The chemical stability of protein crystals has been the subject of intense research,11-14 but our understanding of the mechanical *Corresponding author. E-mail: [email protected]. (1) Malek, K. Biotechnol. Lett. 2007, 29, 1865–1873. (2) Hu, Zh.; Jiang, J. Langmuir 2008, 24, 4215–4223. (3) Eddaoudi, M.; Kim, J.; Rosi, N.; Vodak, D.; Wachter, J.; O’Keefe, M.; Yaghi, O. M. Science 2002, 295, 469. (4) St Clair, N.; Shenoy, B.; Jacob, J. D.; Margolin, A. L. Proc. Natl. Acad. Sci. U.S.A. 1999, 96, 9469–9474. (5) Gosline, J.; Lillie, M.; Carrington, E.; Guerrette, P.; Ortlepp, C.; Savage, K. Philos. Trans. R. Soc. London, Ser. B 2002, 357, 121–132. (6) Speziale, S.; Jiang, F.; Caylor, Y. C. L.; Kriminski, Y. S.; Zha, C.-S.; Thorne, R. E.; Duffy, T. S. Biophys. J. 2003, 85, 3202–3213. (7) Blundell, T. L.; Jhoti, H.; Abell, C. Nat. Rev. Drug Discovery 2003, 1, 45–54. (8) Vilenchik, L. Z.; Griffith, J. P.; St. Clair, N. L.; Navia, M. A.; Margolin, A. L. J. Am. Chem. Soc. 1998, 120, 4290–4294. (9) Margolin, A. L.; Navia, M. A. Angew. Chem., Int. Ed. 2001, 40, 2204–2222. (10) Cvetkovic, A.; Picioreanu, C.; Straathof, A. J. J.; Krishna, R.; Van der Wielen, L. A. M. J. Phys. Chem. B 2005, 109, 10561–10566. (11) Pechenov, S.; Shenoy, B.; Yang, M. X.; Basu, S. K.; Margolin J. Controlled Release 2004, 96, 149–158. (12) Margolin, A. L.; Navia, M. A. Angew. Chem., Int. Ed. 2001, 40, 2204–2222. (13) Shenoy, B.; Wang, Y.-F.; Shan, W.; Margolin, A. L. Biotechnol. Bioeng. 2001, 73, 358–369. (14) Yang, M. X.; Shenoy, B.; Disttler, D.; Patel, R.; McGrath, M.; Pechenov, S.; Margolin, A. L. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 6934–6939.

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stability of protein crystals under different conditions still remains largely obscure. However, the structure, behavior, and mechanical properties of protein crystals play important roles in their performance and life cycle.15,16 Previous studies have shown that protein crystals are relatively fragile and soft materials, and their mechanical properties such as Young’s modulus and hardness are very sensitive to environmental conditions and the type of the protein being studied.16 For example, crystalline glucose isomerase was found to be more ductile and less fragile than crystalline lysozyme in the same environment.16 Although such studies provide valuable qualitative information about the mechanical properties of protein crystals, a detailed quantitative understanding of their anisotropic mechanical response is not possible by experimentation alone as mechanical testing would then be required for many different crystallographic orientations. In this work we introduce a strategy for modeling the anisotropic mechanical response of protein crystals with tetragonal lysozyme as a model. We apply different procedures to explore the orientation dependent mechanical stability of this crystal under external forces and different conditions of temperature and amounts of intracrystalline water. Lysozyme is an enzyme which is found in egg white, tear, saliva, mucus, and other body fluids. Its main role is to damage bacterial cell walls and prevent infection. Lysozyme can be easily crystallized, and its well-known and stable structure makes it a good choice for our study. Some of the mechanical properties of this crystal have been investigated using microhardness measurements and sound velocity under different temperatures and amounts of intracrystalline water.16-20 It has been observed that temperature and amount of intracrystalline water have significant effects on (15) Buehler, M. J. J. Comput. Theor. Nanosci. 2006, 3, 670–683. (16) Tait, S.; White, E. T.; Litster, J. D. Part. Part. Syst. Charact. 2008, 25, 266. (17) Koizumi, H.; Tachibana, M.; Kojima, K. Phys. Rev. E 2006, 73, 041910. (18) Tachibana, M.; Kobayashi, Y.; Shimazu, T.; Ataka, M.; Kojima, K. J. Cryst. Growth 1999, 198/199, 661–664. (19) Koizumi, H.; Tachibana, M.; Kawamoto, H.; Kojima, K. Philos. Mag. 2004, 84, 2961–2968. (20) Koizumi, H.; Kawamoto, H.; Tachibana, M.; Kojima, K. J. Phys. D: Appl. Phys. 2008, 41, 074019.

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Figure 1. (a) A lysozyme molecule (PDB code: 133 L)21 and (b) the

where N is the number of slip systems in the crystal, and γ_ R is the shear slip rate. To evaluate the overall deformation behavior of a single crystal, a single yield function is used in order to calculate the crystal spin and incremental shear strains on the active slip systems. The idea of using a single yield function for crystals was introduced by Montheillet et al.22 and then improved and validated by others.23-30 This model has been widely used to predict the deformation in atomic crystals, but not molecular crystals such as tetragonal lysozyme crystals. Such crystal plasticity model can be also used for molecular crystals as they can have the same deformation mechanisms as atomic crystals.31 To explain such a yield function, let the shear strain rate on any active slip system can be expressed by27  2n -1 sgnðτR Þ τR  R γ_ ¼ λ ð6Þ   τ 0 R τ 0 R 

tetragonal lysozyme crystal with its crystal coordinates and (110) slip plane.

where τR is the resolved shear stress on the slip system which can be related to the stress tensor in the fixed coordinate system by

the elastic and plastic properties of tetragonal lysozyme crystals. At lower temperatures and water content, the crystal is more brittle while it is more ductile at higher temperature and humidity. Dislocations have been found to be the main source of plastic deformation in this crystal.18 In previous works the plastic behavior of lysozyme crystals was explored using microindentation on specific molecular planes.16-20 Although these investigations revealed the plastic deformation mechanisms, they did not provide enough information about the complex anisotropic mechanical behavior. In this work, we use the existing experimental data with a combination of analytical and computational methods to explore the anisotropic mechanical response of lysozyme crystals.

τR ¼ σ : PR

2. Analysis and Methods In this work we have used an experimentally validated micromechanical model to predict the mechanical behavior of tetragonal lysozyme single crystals (Figure 1) under uniaxial loading. To explain the model, let m be a unit vector normal to the slip plane and s a unit vector in the slip direction in the crystal coordinate system. Then, any slip system can be defined by an orientation matrix IR ¼ sR XmR

ð1Þ

with symmetric and antisymmetric parts 1 PR ¼ ðIR þ IRT Þ 2

ð2Þ

1 wR ¼ ðIR -IRT Þ 2

ð3Þ

which define the plastic rate of deformation Dp and spin rate Ωp as Dp ¼

N X R ¼1

Ωp ¼

N X R ¼1

γ_ R 3 PR

ð4Þ

γ_ R 3 wR

ð5Þ

(21) Harata, K.; Muraki, M.; Jigami, Y. J. Mol. Biol. 1993, 233, 524.

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ð7Þ

In eq 6, λ is a positive parameter which depends on the type of dislocation barriers, τ0R is the critical resolved shear stress (CRSS) on the active slip system, and n is a material parameter. Substituting eq 6 into eqs 4 and 5  2n -1 N R  R X sgnðτ Þ τ  R Dp ¼ λ   3P R τ R  τ 0 0 R ¼1

ð8Þ

 2n -1 N X sgnðτR Þ τR  R Ω ¼λ   3w R τ R  τ 0 0 R ¼1

ð9Þ

p

Considering eqs 8 and 9 and using the normality rule in plasticity, one can introduce a plastic potential with respect to the crystal coordinate system as27 0  2n 1 X  τR  1 f ðσ, τ0 R Þ ¼ @  R  -1A τ 0  2n

ð10Þ

The tetragonal lysozyme crystals belongs to the P43212 space group with lattice constants of a = b = 7.91 nm, c = 3.79 nm, and Z = 8.18 The elastic constants of lysozyme crystal are reported to be sensitive to both temperature and humidity.17,19,20,32,33 It has been reported that the Young’s modulus decreases with increasing temperature according to the following relationship:19 ΔE ¼ -CT E0 ΔT

ð11Þ

(22) Montheillet, F.; Gilormini, P.; Jonas, J. J. Acta Metall. 1985, 33, 705. (23) Van Houtte, P. Textures Microstruct. 1987, 7, 29. (24) Lequeu, P. H.; Gilormini, P.; Montheillet, F.; Bacroix, B.; Jonas, J. J. Acta Metall. 1987, 35, 439. (25) Arminjon, M. Textures Microstruct. 1991, 14-18, 1121. (26) Darrieulat, M.; Piot, D. Int. J. Plast. 1996, 12, 575. (27) Gambin, W.; Barlat, F. Int. J. Plast. 1997, 13, 75. (28) Zamiri, A.; Pourboghrat, F.; Barlat, F. Int. J. Plast. 2007, 23, 1126. (29) Zamiri, A.; Pourboghrat, F.; Bieler, T. R. J. Appl. Phys. 2008, 104, 084904. (30) Zamiri, A.; Pourboghrat, F. Int. J. Plast. DOI: 10.1016/j.ijplas.2009.10.004. (31) Alvarado-Contreras, J.; Polak, M. A.; Penlidis, A. Polym. Eng. Sci. 2007, 47, 410. (32) Koizumi, H.; Tachibana, M.; Kojima, K. Phys. Rev. E 2009, 79, 061917. (33) Speziale, S.; Jiang, F.; Caylor, C. L.; Kriminski, S.; Zha, C. -S.; Thorne, R. E.; Duffy, T. S. Biophys. J. 2003, 85, 3202.

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Figure 2. Vickers indenter, A, and its equivalent cone, B. For simulation purposes the indenter A was replaced by indenter B. Table 1. Slip Systems and Corresponding Parameters in Tetragonal Lysozyme Crystals17,20,32,39 slip system (110)[001] (110)[001] (110)[110] (110)[110]

Burgers vector, b (nm) wet: 3.79 dry: 3.12 wet: 5.59 dry: 5.22

Young’s modulus, E (GPa) wet: 2.01 dry: 7.25

self-energy (10-9 J/m) wet: 10.0 dry: 25.6 wet: 21.8 dry: 71.9

where ΔE and ΔT are increments in the Young’s modulus and temperature, respectively, E0 is the Young’s modulus at 300 K, and CT is the temperature coefficient of the Young modulus which is 2  10-3 K-1 for lysozyme crystals.19 It has also been found that the Young’s modulus increases with increase in the amount of the intracrystalline water molecules17,20 according to the relationship ΔE ¼ Cw E0 Δt

ð12Þ

where Δt is the evaporation time and Cw is the humidity coefficient whose value depends on environmental parameters such as temperature. Using the experimental data in ref 20 for natural evaporation of water from lysozyme crystal surface at room temperature, Cw was calculated as 0.0396 (1/min). Lysozyme crystals exhibit elastic-plastic response under external loading.16,17 The plastic deformation mechanism is known to be due to crystallographic slip.18 The crystal has two sets of slip systems:19 a primary {110}Æ001æ system and a secondary {110}Æ110æ system. Some of the important parameters corresponding to these slip systems are shown in Table 1 and are used in our model. In this table, “dry” and “wet” conditions correspond to 42% and 98% relative humidity, respectively. A major parameter appearing in our model is the critical resolved shear stress (τ0R) in eq 10 of the slip systems which is defined as the minimum shear stress to active a slip system. To determine the CRSS for different environmental conditions of temperature and humidity, we have performed computer simulations of microindentation experiments of lysozyme single crystals. This technique of characterizing CRSS from experimental data is well-known and has been employed in other material systems.34-36 During these tests, a diamond indenter is forced to penetrate the specimen under loading. The hardness (HV) of the specimen is expressed as HV = P/A, where P is the maximum load applied during indentation and A is the projected contact area of (34) Sheyka, M.; El-Kady, I.; Khraishi, T.; Reda Taha, M. M. Int. J. Mech. Mater. Des. 2008, 4, 407. (35) Li, X.; An, Y. H.; Wu, Y-D.; Song, Y. C.; Chao, Y. J.; Chien, C. H. J. Biomed. Mater. Res., Part B: Appl. Biomater. 2007, 80B, 25. (36) Zhang, J.; Niebur, J. L.; Ovaert, T. C. J. Biomech. 2008, 41, 267.

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Figure 3. Finite element model of a Vickers microindentation test used in our analysis (R = 100 μm, t = 120 μm, req = 55.86 μm, and h = 20 μm).

the indenter. In the Vickers microhardness measurement technique, the indenter is a pyramid and the hardness is expressed as HV = 1.854(P/d2), where d is the diagonal of the indenter (Figure 2). In our simulation of Vickers microindentation experiments, the geometry of the Vickers indenter was chosen according to the procedure proposed in refs 37 and 38. To avoid singularity and to reduce computational ambiguity, the Vickers pyramid was modeled as an equivalent cone (Figure 2). Since only the projected area of the indenter enters into the computation of the hardness, the equivalent radius of √ the cone at an indentation depth of h is computed as req = a/ π, where a is the side length of the Vickers pyramid corresponding to the same depth (Figure 2). This ensures that the cross-sectional areas A and B of the pyramid and conical indenters at the same penetration depth are equal. In order to account for anisotropy, a cylindrical finite element model of a lysozyme crystal was developed (Figure 3) which was fixed at the bottom surface. At the Gauss integration point of all finite elements in the model, a unit cell composed of a lysozyme crystal which its (110) molecular plane is parallel to the radius of the cylindrical model was chosen. The contact between the indenter and the specimen was assumed to be frictionless. This model was then used to simulate the micro-Vickers hardness tests on the (110) plane to obtain the force and tip displacement curves at different temperatures and amounts of intracrystalline water. The same boundary conditions that were used in refs 17-20 were applied here. The simulations were conducted at six different temperatures between 278 and 307 K with indentation load of 9.8 mN and at different natural evaporation times between 0 and 100 min at room temperature with indentation load of 9.8, 19.6, and 49 mN for 0-10, 10-30, and 30-100 min of evaporation times, respectively. Having the hardness values HV and the indentation forces P from experiments, the diagonals of indentation d were computed using the equation HV = 1.854(P/d2) for different (37) Bouzakisa, K. D.; Michailidisa, M.; Erkensb, G. Surf. Coat. Technol. 2001, 142/144, 102–109. (38) Shih, C. W.; Yong, M.; Li, J. C. M. J. Mater. Res. 1991, 6, 2623.

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Figure 4. Force versus indentation depth curves corresponding to microindentation simulations under three different cases: dry condition, wet condition at 298 K, and wet condition at 307 K.

values of temperature and humidity. Then the side lengths of the Vickers pyramid a were calculated as a = 21/2d/2. The depths of indentations were obtained by h = a/(2 tan 68), where 68 is the Vickers indenter angle. After computing the depth of indentation h for different temperatures and humidity, the simulation of indentations of lysozyme crystals were carried out up to the indentation depth of h to find CRSS for lysozyme. These simulations were used to determine the CRSS values for the different slip sets such that the forces obtained from the simulations matched published experimental values. After obtaining the CRSS values as discussed in the previous section, uniaxial compression of the lysozyme crystals was carried out on a 8-noded brick finite element with unit sides using the commercial software code Abaqus at a rate of 0.05 s-1 to explore the anisotropic mechanical properties. Associated with each of the eight integration points of the element is a single crystal of lysozyme which is oriented with a specific orientation with respect to the lab coordinate system. During the simulation, the two faces of the cube normal to the axis of loading were constrained such that they remained parallel to each other throughout the simulation. The results of these simulations are discussed in the next section.

3. Results and Discussion Figure 4 shows the results of the microindentation tests on a lysozyme single crystal on the (110) molecular plane in the wet condition at 298 and 307 K and in the dry condition at room temperature. It is clear that lysozyme becomes softer with increasing temperature and water content. Both temperature and intracrystalline water affect the elastic and plastic deformation regimes. The effects of temperature and pressure are accounted for in the elastic constants (eqs 11 and 12) and in the CRSS (τ0R) of the slip systems (eq 10). In the following subsections, we explore the effects of temperature and intracrytalline water on the mechanical response of lysozyme crystals through their effects on the CRSS of slip systems. 3.1. Effect of Temperature on Yield Stress. Figure 5 shows a plot of the hardness and yield stress for a lysozyme crystal indented on its (110) molecular plane in the temperature range of 278-307 K. The CRSS is computed from hardnesses values extracted from ref 19 as described in section 2. The numerical compression tests are then performed to compute the yield stresses. The yield stress decreases with increasing temperature. However, it is less temperature sensitive at temperatures below room temperature. This may be explained based on the understanding that the deformation mechanism of lysozyme crystals is (39) Tachibana, M.; Koizumi, H.; Kojima, K. Phys. Rev. E 2004, 69, 051921.

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Figure 5. Yield stress and hardness at different temperatures along [110] crystallographic direction (the hardness values were extracted from ref 19 for comparison).

Figure 6. Yield stress and hardness at different states of hydration along the [110] crystallographic direction (the hardness values were extracted from 20 for comparison).

predominantly elastic below room temperature but is elasticplastic at higher temperatures.19 Since the plastic deformation, which is due to the dislocation activation, is more temperature sensitive than the elastic deformation, the slope of the yield stress curve increases at higher temperatures. This will be discussed in more detail in the following sections. 3.2. Effect of Intracrystalline Water on Yield Stress. The amount of intercrystalline water molecules, which is a function of the evaporation time, has a significant effect on the mechanical properties of protein crystals. The hardness decreases with increasing water content. Figure 6 shows the effect of the evaporation time on the yield stress. The hardness values are extracted from ref 20 and used to compute the CRSS. With evaporation, the number of water molecules in the lattice decreases and this increases the yield stress. Three different regimes of interest may be identified in this graph. For the first 20 min of evaporation time (stage 1) the yield stress remains almost constant at about 5.8 MPa. Between 20 and 40 min of evaporation (stage 2) the yield stress increases to about 45 MPa, and beyond that (stage 3) the yield stress continues to increase but with a different slope. On the basis of our microindentation analysis, we hypothesize that this may be related to the activation of the slip systems in the three different regimes. In stage 1 both slip system sets (Table 1) are easily activated while in stage 2 the Langmuir 2010, 26(6), 4251–4257

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Figure 7. Effect of temperature on critical resolved shear stresses. Figure 8. Effect of the amount of intracrystalline water molecule on critical resolved shear stresses of slip systems in lysozyme crystal.

effect of the secondary slip systems fades gradually and it is no longer active in stage 3. As mentioned earlier, the yield stress of a crystal has a direct relation to its CRSS values. In the following sections we discuss how the temperature and intracrystalline water affect the CRSS values in tetragonal lysozyme crystals. 3.3. Temperature Dependence of the Critical Resolved Shear Stress (CRSS). On the basis of our microindentation simulations, we plot the CRSS for the two different slip systems of the lysozyme crystal at different temperatures in Figure 7. This is in agreement with the experimental observation that the selfenergy of dislocations for the {110}Æ001æ slip system is larger than that for the {110}Æ110æ slip system.19 The self-energy of dislocations is a function of the shear modulus, G, and the magnitude of the Burgers vector, b, and is proportional to Gb2. As in eq 1, the Young’s modulus of the lysozyme crystal decreases with increasing temperature and so does the shear modulus, which is related to the Young’s modulus according to the relationship E = 2G(1 þ ν), where ν is the Poisson’s ratio. Thus, the self-energy of the dislocations in lysozyme crystal decreases with increasing temperature which eases the nucleation of dislocations, thereby decreasing the CRSS. However, it is reported in ref 19 that the effect of temperature on the hardness of lysozyme crystals is more significant than its effect on the elastic constants. Therefore, we may conclude that the effect of temperature on the CRSS must also be related to the dislocation mechanisms. As mentioned before, plastic flow in lysozyme crystals occurs by creation and motion of dislocations. It is recognized that thermal fluctuations provide the energy to carry the dislocations over the lattice potential barriers.40,41 Hence, dislocations at higher temperatures have a higher probability of overcoming lattice potential barriers due to higher thermal fluctuations. This explains why the CRSS decreases with increasing temperature for both slip systems, as shown in Figure 7. At higher temperatures, the CRSS of the two slip systems are smaller and closer to each other, and therefore, both the slip systems can be easily activated. At lower temperatures, the CRSS of the {110}Æ001æ slip system is much less than the CRSS of the {110}Æ110æ slip system and can therefore be more easily activated. This difference in slip activation behavior is the cause of the anisotropic deformation characteristics at different temperatures. As discussed before, at temperatures below room temperature, the deformation in lysozyme crystals is primarily elastic while at higher temperatures it is elastic-plastic. Therefore, at lower temperatures the temperature variation of the yield stress in Figure 5 is primarily due to the temperature dependence of the

elastic constant and hence the CRSS. However, at higher temperatures, both the elastic constant and dislocation mechanisms are affected by temperature which results in a higher drop in yield stress with increasing temperature. 3.4. Effect of Intracrytalline Water on CRSS. Protein molecules have a significant amount of intercrystalline water. The decrease in CRSS with increasing temperature (Figure 7) may potentially be related to these water molecules. Two types of intracrystalline water may be present in the lattice: mobile water, which can easily traverse through the crystal, and bounded water, which is bound to the molecules.19,20 Mobile water has a high diffusion coefficient at higher temperatures42 and therefore has little interaction with dislocations. However, at lower temperatures it may interact with dislocations and thereby affect dislocation creation and motion in the lattice.19 As shown in Figure 8, the CRSS for both slip system sets increases with evaporation time. This can be explained as follows. It has been reported that a change in intracrystalline water changes both the lattice and elastic constants of protein crystals. A decrease in the amount of the intracrystalline water, as a result of evaporation, leads to an increase in elastic constants and a decrease in lattice parameters.39 This increases the self-energy of the dislocations significantly and hinders their nucleation and activation, thereby increasing the CRSS of the slip systems. Another parameter that may significantly affect the CRSS is the Peierls stress. The Peierls stress is the minimum force necessary to move a dislocation within an atomic/molecular plane in a crystal. It is an intrinsic property of a crystal and depends on temperature and impurity in the crystal. It has been observed that the Peierls stress of lysozyme crystals significantly increases with decreasing amounts of intracrystalline water.19 One hypothesis based on molecular analysis suggests that with evaporation the lattice constants decrease; therefore, the number of atom-atom interactions on the surface of the protein molecules increases, and this enhances the intermolecular interaction and thus the Peierls stress.20,43 However, a detailed understanding of the Peierls stress in protein crystals is yet to be achieved. This is due to the difficulty of studying the interactions of adjacent molecular planes when the lattice sites are occupied by complex protein molecules. In this study only the effects of the water molecules was investigated. In some applications, e.g., drug delivery, protein crystals contain drug molecules or other fluid molecules in the lattice. The presence of these molecules can have more complex

(40) Kocks, U. F.; Argon, A. S.; Ashby, M. F. In Progress in Materials Science; Chalmers, B., Christian, J. W., Massalski, T. B., Eds.; Pergamon: New York, 1975. (41) Granto, A. V.; Lucke, K.; Schlipf, J.; Teutonico, L. J. J. Appl. Phys. 1964, 35, 2732.

(42) Morozov, V. N.; Kachalova, G. S.; Evtodienko, V. U.; Lanina, N. F.; Morozova, T. Y. Eur. Biophys. J. 1995, 24, 93. (43) Matsuura, Y.; Chernov, A. A. Acta Crystallogr., Sect. D: Biol. Crystallogr. 2003, 59, 1347.

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Zamiri and De Table 2. Schmid Factors Computed for the Slip Systems of Tetragonal Lysozyme Crystal for Different Loading Directions slip systems initial loding direction [100] [110] [212]

Figure 9. Stress-strain response of a lysozyme crystal along different crystallographic directions: (a) in wet condition at room temperature, (b) in wet condition at 307 K, and (c) in dry condition at room temperature.

effects on the mechanical properties which have to be studied separately. 3.5. Anisotropic Mechanical Response of Lysozyme Crystals. To evaluate the mechanical response of lysozyme single crystals under different conditions, compression tests were performed along the different crystallographic directions for dry and wet crystals at two different temperatures. Two of the crystallographic loading directions were chosen in such a way that in one only the (110)[001] type slip systems become activated ([212] loading direction) and in the other only the (110)[110] type slip systems get activated ([110] loading direction). Such selection of loading directions provides valuable insight into the effect of different slip systems on the mechanical response of the lysozyme crystal. Figure 9 shows the stress-strain curves for different orientations at three different conditions. The deformation behavior is observed to be highly anisotropic. The wet crystals loaded along the [100] and [212] directions exhibit elastic-plastic response whereas compression along the [110] direction results in purely elastic response (Figure 9a,b). The yield stress along [212] direc4256 DOI: 10.1021/la9033222

(110)[001]

(110)[001]

(110)[110]

(110)[110]

0.0 0.0 0.42

0.0 0.0 0.09

0.50 0.0 0.16

0.50 0.0 0.16

tion is lower compared to the [100] direction. When the wet crystals are loaded along the [212] direction, the (110)[001] slip system is activated first followed by the (110)[001] slip system. As is clear from the figure, a small hardening appears when loaded along [212]. Since the plastic part of the deformation is very small in lysozyme crystals and, therefore, the crystal is considered as elastic-perfectly plastic, this hardening is due to the lattice rotation with plastic deformation. The [212] direction is not symmetric with respect to the loading direction, and hence, the lattice rotates during deformation from [212] to an orientation that is less favorable for crystal slip which results in the hardening. Such behavior has been also observed in other single crystals.44 Loading along the [100] direction activates both (110)[110] and (110)[110] slip systems at the same time. The dry crystal shows purely elastic deformation along both [100] and [110] crystallographic directions (Figure 9c) whereas its response is elastic-plastic along the [212] direction. In this case also the (110)[001] slip system is activated first followed by the activation of the (110)[001] slip system. Table 2 shows the Schmid factors (defined as the ratio of the resolved shear stress, τR, and the uniaxial stress) calculated for all slip systems for the above crystal orientations. The Schmid factors clearly show that loading along the [100] direction leads to simultaneous activation of both (110)[110] and (110)[110] slip systems since they have the greatest Schmid factors. For [110] orientation no slip system is available since for all slip systems the Schmid factor is 0.0. Therefore, it remains elastic during uniaxial compression. For the [212] orientation, the highest Schmid factor is for the (110)[001] slip system which is the first to be activated. As the plastic part of the deformation is very small in lysozyme crystals, it was treated as elastic-perfectly plastic in our analyses. However, the material may show some strain hardening (or softening) during the plastic deformation. To account for this in the current model, the evolution of the CRSS, τ0R, with respect to the strain, strain rate, humidity, and temperature must be clearly understood. For this purpose, more experiments or molecular level simulations including compression tests and indentations at different scale and environmental conditions are necessary. Previous molecular dynamic simulations show that deformation in protein crystals is also largely dependent on the deformation and fractures of the protein molecules15 which may lead to heterogeneous and orientation-dependent deformation in the lattice. As opposed to the atomic and ionic crystals, protein crystals are composed of large flexible molecules with irregular shapes in the lattice points that make the deformation and mechanical response of such crystals much more complex at the molecular level. Another very important factor controlling the mechanical response of protein crystals is creation, motion, and interaction of defects such as dislocations.18 These initiation and interactions of defects could be incorporated into the current model through development of sophisticated physical based models for the evolution of the CRSS with strain. Such extensions of the model presented in this paper would enable the prediction (44) Anand, L.; Kothari, M. J. Mech. Phys. Solids 1996, 44, 525.

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Zamiri and De

Article

of dislocation and defect related damage and microcrack initiation and propagation. Once developed, such a model will have advantage over atomistic models as it would be computationally much more efficient and would be capable of simulating much larger spatial and temporal scales. It is also possible to link the continuum micromechanical approach proposed here with a molecular level computational model to not only take into account the defect interactions at larger scales but also to account for molecular flexibility and bond breaking at the molecular scale. Such a hybrid continuumatomistic approach could provide much valuable insight into the complex deformation mechanics of biomolecular crystals as well as critical biological processes such as mutations under different macroscopic environmental and loading conditions.

4. Conclusion The mechanical response of the tetragonal lysozyme crystals has been investigated based on a combination of indentation simulations, crystal plasticity modeling, and existent experiments. The results of our investigation show that the mechanical strength of the lysozyme crystal is significantly sensitive to temperature and the amount of intracrystalline water molecules. The critical

Langmuir 2010, 26(6), 4251–4257

resolved shear stresses for all slip systems decreases with increasing temperature and the quantity of intracrystalline water molecules. Further analysis of the deformation along different crystallographic directions shows that the deformation and the yield stress are highly anisotropic with certain directions exhibiting purely elastic response, whereas others exhibiting elastoplastic behavior. The analysis presented here may serve as a framework for the investigation of the mechanical response of various protein crystals. This may also be used to develop optimal design strategies for bionanoporous materials for future applications. However, further work is necessary, especially in experimental characterization of protein crystals, to develop sophisticated models for postyield behavior, hardening, damage, and softening under different environmental conditions. When the intracrystalline spaces are filled by fluids other than water, or other complex molecules, specialized analysis may be necessary. Acknowledgment. We gratefully acknowledge the support of this work through Office of Naval Research grants N000140510686 and N000140810462 with Dr. Clifford Bradford as the cognizant Program Manager.

DOI: 10.1021/la9033222

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