0
Research Papers

Displacement Propagation in Fibrous Networks Due to Local Contraction

[+] Author and Article Information
Peter Grimmer

Department of Engineering Physics,
University of Wisconsin-Madison,
1500 Engineering Drive,
Madison, WI 53706
e-mail: pgrimmer@wisc.edu

Jacob Notbohm

Department of Engineering Physics,
University of Wisconsin-Madison,
1500 Engineering Drive,
Madison, WI 53706
e-mail: jknotbohm@wisc.edu

1Corresponding author.

Manuscript received June 19, 2017; final manuscript received November 19, 2017; published online February 12, 2018. Assoc. Editor: Thao (Vicky) Nguyen.

J Biomech Eng 140(4), 041011 (Feb 12, 2018) (11 pages) Paper No: BIO-17-1267; doi: 10.1115/1.4038744 History: Received June 19, 2017; Revised November 19, 2017

The extracellular matrix provides macroscale structure to tissues and microscale guidance for cell contraction, adhesion, and migration. The matrix is composed of a network of fibers, which each deform by stretching, bending, and buckling. Whereas the mechanics has been well characterized in uniform shear and extension, the response to more general loading conditions remains less clear, because the associated displacement fields cannot be predicted a priori. Studies simulating contraction, such as due to a cell, have observed displacements that propagate over a long range, suggesting mechanisms such as reorientation of fibers toward directions of tensile force and nonlinearity due to buckling of fibers under compression. It remains unclear which of these two mechanisms produces the long-range displacements and how properties like fiber bending stiffness and fiber length affect the displacement field. Here, we simulate contraction of an inclusion within a fibrous network and fit the resulting radial displacements to ur ∼ rn where the power n quantifies the decay of displacements over distance, and a value of n less than that predicted by classical linear elasticity indicates displacements that propagate over a long range. We observed displacements to propagate over a longer range for greater contraction of the inclusion, for networks having longer fibers, and for networks with lower fiber bending stiffness. Contraction of the inclusion also caused fibers to reorient into the radial direction, but, surprisingly, the reorientation was minimally affected by bending stiffness. We conclude that both reorientation and nonlinearity are responsible for the long-range displacements.

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Discher, D. E. , Janmey, P. , and Wang, Y.-L. , 2005, “ Tissue Cells Feel and Respond to the Stiffness of Their Substrate,” Science, 310(5751), pp. 1139–1143. [CrossRef] [PubMed]
Riching, K. M. , Cox, B. L. , Salick, M. R. , Pehlke, C. , Riching, A. S. , Ponik, S. M. , Bass, B. R. , Crone, W. C. , Jiang, Y. , Weaver, A. M. , Eliceiri, K. W. , and Keely, P. J. , 2014, “ 3D Collagen Alignment Limits Protrusions to Enhance Breast Cancer Cell Persistence,” Biophys. J., 107(11), pp. 2546–2558. [CrossRef] [PubMed]
Provenzano, P. P. , Eliceiri, K. W. , Campbell, J. M. , Inman, D. R. , White, J. G. , and Keely, P. J. , 2006, “ Collagen Reorganization at the Tumor-Stromal Interface Facilitates Local Invasion,” BMC Med., 4(1), p. 38. [CrossRef] [PubMed]
Conklin, M. W. , Eickhoff, J. C. , Riching, K. M. , Pehlke, C. A. , Eliceiri, K. W. , Provenzano, P. P. , Friedl, A. , and Keely, P. J. , 2011, “ Aligned Collagen Is a Prognostic Signature for Survival in Human Breast Carcinoma,” Am. J. Pathol., 178(3), pp. 1221–1232. [CrossRef] [PubMed]
Ye, Q. , Zünd, G. , Benedikt, P. , Jockenhoevel, S. , Hoerstrup, S. P. , Sakyama, S. , Hubbell, J. A. , and Turina, M. , 2000, “ Fibrin Gel as a Three Dimensional Matrix in Cardiovascular Tissue Engineering,” Eur. J. Cardiothorac. Surg., 17(5), pp. 587–591. [CrossRef] [PubMed]
Boccafoschi, F. , Habermehl, J. , Vesentini, S. , and Mantovani, D. , 2005, “ Biological Performances of Collagen-Based Scaffolds for Vascular Tissue Engineering,” Biomaterials, 26(35), pp. 7410–7417. [CrossRef] [PubMed]
Licup, A. J. , Munster, S. , Sharma, A. , Sheinman, M. , Jawerth, L. M. , Fabry, B. , Weitz, D. A. , and MacKintosh, F. C. , 2015, “ Stress Controls the Mechanics of Collagen Networks,” Proc. Natl. Acad. Sci. U. S. A., 112(31), pp. 9573–9578. [CrossRef] [PubMed]
Sharma, A. , Licup, A. J. , Jansen, K. A. , Rens, R. , Sheinman, M. , Koenderink, G. H. , and MacKintosh, F. C. , 2016, “ Strain-Controlled Criticality Governs the Nonlinear Mechanics of Fibre Networks,” Nat. Phys., 12(6), pp. 584–587. [CrossRef]
Vahabi, M. , Sharma, A. , Licup, A. J. , van Oosten, A. S. , Galie, P. A. , Janmey, P. A. , and MacKintosh, F. C. , 2016, “ Elasticity of Fibrous Networks Under Uniaxial Prestress,” Soft Matter, 12(22), pp. 5050–5060. [CrossRef] [PubMed]
Head, D. A. , Levine, A. J. , and MacKintosh, F. C. , 2003, “ Distinct Regimes of Elastic Response and Deformation Modes of Cross-Linked Cytoskeletal and Semiflexible Polymer Networks,” Phys. Rev. E, 68(6), p. 061907. [CrossRef]
Wilhelm, J. , and Frey, E. , 2003, “ Elasticity of Stiff Polymer Networks,” Phys. Rev. Lett., 91(10), p. 108103. [CrossRef] [PubMed]
Onck, P. R. , Koeman, T. , van Dillen, T. , and van der Giessen, E. , 2005, “ Alternative Explanation of Stiffening in Cross-Linked Semiflexible Networks,” Phys. Rev. Lett., 95(17), p. 178102. [CrossRef] [PubMed]
Heussinger, C. , and Frey, E. , 2007, “ Force Distributions and Force Chains in Random Stiff Fiber Networks,” Eur. Phys. J. E, 24(1), pp. 47–53. [CrossRef]
Hatami-Marbini, H. , and Picu, R. C. , 2008, “ Scaling of Nonaffine Deformation in Random Semiflexible Fiber Networks,” Phys. Rev. E, 77(6), p. 062103. [CrossRef]
Stein, A. M. , Vader, D. A. , Weitz, D. A. , and Sander, L. M. , 2011, “ The Micromechanics of Three-Dimensional Collagen-I Gels,” Complexity, 16(4), pp. 22–28. [CrossRef]
Fung, Y. C. , 2013, Biomechanics: Mechanical Properties of Living Tissues, Springer Science & Business Media, New York.
Janmey, P. A. , McCormick, M. E. , Rammensee, S. , Leight, J. L. , Georges, P. C. , and MacKintosh, F. C. , 2007, “ Negative Normal Stress in Semiflexible Biopolymer Gels,” Nat. Mater., 6(1), pp. 48–51. [CrossRef] [PubMed]
Conti, E. , and MacKintosh, F. , 2009, “ Cross-Linked Networks of Stiff Filaments Exhibit Negative Normal Stress,” Phys. Rev. Lett., 102(8), p. 088102. https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.102.088102
Poynting, J. H. , 1909, “ On Pressure Perpendicular to the Shear Planes in Finite Pure Shears, and on the Lengthening of Loaded Wires When Twisted,” Proc. R. Soc. London, 82(557), pp. 546–559. [CrossRef]
Notbohm, J. , Lesman, A. , Rosakis, P. , Tirrell, D. A. , and Ravichandran, G. , 2015, “ Microbuckling of Fibrin Provides a Mechanism for Cell Mechanosensing,” J. R. Soc. Interface, 12(108), p. 20150320. [CrossRef] [PubMed]
Rosakis, P. , Notbohm, J. , and Ravichandran, G. , 2015, “ A Model for Compression-Weakening Materials and the Elastic Fields Due to Contractile Cells,” J. Mech. Phys. Solids, 85, pp. 16–32. [CrossRef]
Kim, O. V. , Litvinov, R. I. , Weisel, J. W. , and Alber, M. S. , 2014, “ Structural Basis for the Nonlinear Mechanics of Fibrin Networks Under Compression,” Biomaterials, 35(25), pp. 6739–6749. [CrossRef] [PubMed]
van Oosten, A. S. , Vahabi, M. , Licup, A. J. , Sharma, A. , Galie, P. A. , MacKintosh, F. C. , and Janmey, P. A. , 2016, “ Uncoupling Shear and Uniaxial Elastic Moduli of Semiflexible Biopolymer Networks: Compression-Softening and Stretch-Stiffening,” Sci. Rep., 6, p. 19270. [CrossRef] [PubMed]
Liang, L. , Jones, C. , Chen, S. , Sun, B. , and Jiao, Y. , 2016, “ Heterogeneous Force Network in 3D Cellularized Collagen Networks,” Phys. Biol., 13(6), p. 066001. [CrossRef] [PubMed]
Rudnicki, M. S. , Cirka, H. A. , Aghvami, M. , Sander, E. A. , Wen, Q. , and Billiar, K. L. , 2013, “ Nonlinear Strain Stiffening Is Not Sufficient to Explain How Far Cells Can Feel on Fibrous Protein Gels,” Biophys. J., 105(1), pp. 11–20. [CrossRef] [PubMed]
Aghvami, M. , Billiar, K. L. , and Sander, E. A. , 2016, “ Fiber Network Models Predict Enhanced Cell Mechanosensing on Fibrous Gels,” ASME J. Biomech. Eng., 138(10), p. 101006. [CrossRef]
Wang, H. , Abhilash, A. S. , Chen, C. S. , Wells, R. G. , and Shenoy, V. B. , 2014, “ Long-Range Force Transmission in Fibrous Matrices Enabled by Tension-Driven Alignment of Fibers,” Biophys. J., 107(11), pp. 2592–2603. [CrossRef] [PubMed]
Abhilash, A. S. , Baker, B. M. , Trappmann, B. , Chen, C. S. , and Shenoy, V. B. , 2014, “ Remodeling of Fibrous Extracellular Matrices by Contractile Cells: Predictions From Discrete Fiber Network Simulations,” Biophys. J., 107(8), pp. 1829–1840. [CrossRef] [PubMed]
Winer, J. P. , Oake, S. , and Janmey, P. A. , 2009, “ Non-Linear Elasticity of Extracellular Matrices Enables Contractile Cells to Communicate Local Position and Orientation,” PloS One, 4(7), p. e6382. [CrossRef] [PubMed]
Burkel, B. , and Notbohm, J. , 2017, “ Mechanical Response of Collagen Networks to Nonuniform Microscale Loads,” Soft Matter, 13(34), p. 5749. [CrossRef] [PubMed]
Ronceray, P. , Broedersz, C. P. , and Lenz, M. , 2016, “ Fiber Networks Amplify Active Stress,” Proc. Natl. Acad. Sci., 113(11), pp. 2827–2832. http://www.pnas.org/content/113/11/2827.abstract
Eshelby, J. D. , 1959, “ The Elastic Field Outside an Ellipsoidal Inclusion,” Proc. R. Soc. London A, 252(1271), pp. 561–569. [CrossRef]
Lindstrom, S. B. , Vader, D. A. , Kulachenko, A. , and Weitz, D. A. , 2010, “ Biopolymer Network Geometries: Characterization, Regeneration, and Elastic Properties,” Phys. Rev. E, 82(5), p. 051905. [CrossRef]
Lindström, S. B. , Kulachenko, A. , Jawerth, L. M. , and Vader, D. A. , 2013, “ Finite-Strain, Finite-Size Mechanics of Rigidly Cross-Linked Biopolymer Networks,” Soft Matter, 9(30), p. 7302. [CrossRef]
Xu, X. , and Safran, S. A. , 2015, “ Nonlinearities of Biopolymer Gels Increase the Range of Force Transmission,” Phys. Rev. E, 92(3), p. 032728. [CrossRef]
Head, D. A. , Levine, A. J. , and MacKintosh, F. C. , 2005, “ Mechanical Response of Semiflexible Networks to Localized Perturbations,” Phys. Rev. E, 72(6), p. 061914. [CrossRef]
Kang, H. , Wen, Q. , Janmey, P. A. , Tang, J. X. , Conti, E. , and MacKintosh, F. C. , 2009, “ Nonlinear Elasticity of Stiff Filament Networks: Strain Stiffening, Negative Normal Stress, and Filament Alignment in Fibrin Gels,” J. Phys. Chem. B, 113(12), pp. 3799–3805. [CrossRef] [PubMed]
Broedersz, C. P. , Mao, X. M. , Lubensky, T. C. , and MacKintosh, F. C. , 2011, “ Criticality and Isostaticity in Fibre Networks,” Nat. Phys., 7(12), pp. 983–988. [CrossRef]
Broedersz, C. P. , Sheinman, M. , and MacKintosh, F. C. , 2012, “ Filament-Length-Controlled Elasticity in 3D Fiber Networks,” Phys. Rev. Lett., 108(7), p. 078102. [CrossRef] [PubMed]
Münster, S. , Jawerth, L. M. , Leslie, B. A. , Weitz, J. I. , Fabry, B. , and Weitz, D. A. , 2013, “ Strain History Dependence of the Nonlinear Stress Response of Fibrin and Collagen Networks,” Proc. Natl. Acad. Sci., 110(30), pp. 12197–12202. [CrossRef]
Humphries, D. L. , Grogan, J. A. , and Gaffney, E. A. , 2017, “ Mechanical Cell–Cell Communication in Fibrous Networks: The Importance of Network Geometry,” Bull. Math. Biol., 79(3), pp. 498–524. [CrossRef] [PubMed]
Jones, C. A. , Cibula, M. , Feng, J. , Krnacik, E. A. , McIntyre, D. H. , Levine, H. , and Sun, B. , 2015, “ Micromechanics of Cellularized Biopolymer Networks,” Proc. Natl. Acad. Sci. U. S. A., 112(37), pp. E5117–E5122. [CrossRef] [PubMed]
Wood, G. , and Keech, M. K. , 1960, “ The Formation of Fibrils From Collagen Solutions—1: The Effect of Experimental Conditions: Kinetic and Electron-Microscope Studies,” Biochem. J., 75(3), p. 588. [CrossRef] [PubMed]
Shokef, Y. , and Safran, S. A. , 2012, “ Scaling Laws for the Response of Nonlinear Elastic Media With Implications for Cell Mechanics,” Phys. Rev. Lett., 108(17), p. 178103. [CrossRef] [PubMed]
Hatami-Marbini, H. , and Picu, R. C. , 2009, “ Heterogeneous Long-Range Correlated Deformation of Semiflexible Random Fiber Networks,” Phys. Rev. E, 80(4), p. 046703. [CrossRef]
Chandran, P. L. , and Barocas, V. H. , 2006, “ Affine Versus Non-Affine Fibril Kinematics in Collagen Networks: Theoretical Studies of Network Behavior,” ASME J. Biomech. Eng., 128(2), pp. 259–270. [CrossRef]
Kim, O. V. , Liang, X. , Litvinov, R. I. , Weisel, J. W. , Alber, M. S. , and Purohit, P. K. , 2016, “ Foam-Like Compression Behavior of Fibrin Networks,” Biomech. Model. Mechanobiol., 15(1), pp. 213–228. [CrossRef] [PubMed]
Ma, X. , Schickel, M. E. , Stevenson, M. D. , Sarang-Sieminski, A. L. , Gooch, K. J. , Ghadiali, S. N. , and Hart, R. T. , 2013, “ Fibers in the Extracellular Matrix Enable Long-Range Stress Transmission Between Cells,” Biophys. J., 104(7), pp. 1410–1418. [CrossRef] [PubMed]

Figures

Grahic Jump Location
Fig. 1

Fiber network architecture. (a) Domain simulated. The domain's radius is 20 inclusion radii; networks are generated with fibers having average length of 0.27, 0.20, or 0.15 inclusion radii (0.27 shown here). (b) Confocal images of a collagen network. At fiber connections (circles), a single fiber branches into multiple others at acute angles. (c) Two different types of network architecture. Annealed networks are precursors to branched networks. (d) Connectivity distribution for annealed and branched networks. The mean connectivity is 3.3. (e) Branched networks are obtained by reducing the branch angles within the annealed networks. (f) Histograms of fiber lengths and branch angles for the different networks.

Grahic Jump Location
Fig. 9

3D model of contracting spherical inclusion. (a) Undeformed 3D domain, with a horizontal midplane section highlighted. The domain's radius is 8 inclusion radii and the mean fiber length is 0.20 inclusion radii. (b) Themodel is solved using all fibers in panel a; the displacements shown correspond to the midplane section at 32% contraction of the spherical inclusion. (c) Displacement decay rate n. In three dimensions, the analytical solution from classical elasticity gives n = 2, whereas here we see n decreasing as the magnitude of contraction increases. (d) Nonaffinity for the 3D networks. For panels c and d, the different markers correspond to each of four different networks.

Grahic Jump Location
Fig. 8

Results from both contraction and expansion for κ = 10−4 and L = 0.20: (a) displacement field resulting from an expanding inclusion, (b) displacement field in the same network for a contracting inclusion, and (c) displacement decay rate n for both expansion and contraction. For small contraction or expansion n ≈ 1, with higher expansion or contraction giving a greater or smaller decay rate as compared to a classically linear material. Markers show medians; error bars show interquartile range. (d) Nonaffinity for the eight networks for expansion and contraction. Note the logarithmic vertical axis. For panels c and d, the different markers correspond to each of the eight different networks.

Grahic Jump Location
Fig. 2

Displacement field for a network with fiber length L = 0.27 inclusion radii and fiber bending stiffness κ = 10−4. (a) Magnitude of displacements due to radially inward contraction of a circular inclusion by 32%. (b) Radial component of the displacements versus the radial position. The radial displacements are fit to Eq. (1) (solid line). The best fit is n = 0.52, indicating displacements propagate over a longer range than predicted by linear elasticity. For comparison, the 2D linear elastic solution n = 1 is shown by the dashed line. (c) When plotted on logarithmic axes, the fitted value of n matches the slope of the displacements in the regime near the center of the domain, indicating the power n quantifies the rate at which displacements decay over distance. (d) Nonaffine displacements over distance (dots) are fit to the right-hand side of Eq. (1) giving n = 0.46 (solid line). Inset shows the same data on semilogarithmic axes with a fit to a decaying exponential (dash-dot line). (e) Dimensionless nonaffinity χ at each node. Results are shown for only the positions within 75% of the outer radius (shown by thesolid circle in panel a), where the displacements are minimally affected by the boundary.

Grahic Jump Location
Fig. 3

Magnitude of contraction affects propagation of displacements and nonaffinity. (a) The displacement decay rate n versus contraction of the inclusion for networks with fiber length L = 0.27 inclusion radii and fiber bending stiffness κ = 10−4. As contraction increases, the displacements propagate over longer range (decay rate n decreases). (b) Average nonaffinity χ̃. As contraction increases the displacements become more affine with less variation across the networks. For both panels, the different markers correspond to each of eight different random networks having branched geometry.

Grahic Jump Location
Fig. 4

Comparison of different network types. (a) Displacement decay rate n for the two network generation methods, with fiber bending stiffness κ = 10−4. Markers show medians of eight different random networks of each type; error bars give interquartile range. (b) and (c) Network at 32% contraction with fibers above 1% tensile strain darkened. The dots indicate the nodes within the inclusion radius that the displacement boundary condition is applied to. Highly tensed fibers form aligned force chains. A given annealed network (b) exhibits fewer force chains than the associated branched network (c), because the reduced branch angles give pre-aligned load paths for force chains to develop.

Grahic Jump Location
Fig. 5

Effect of fiber length: (a) networks with three different fiber lengths, 0.27, 0.20, and 0.15 inclusion radii, are pictured from left to right, (b) displacement decay rate n for the three fiber lengths, using a fiber bending stiffness of κ = 10−4, and (c) average nonaffinity χ̃ for the three fiber lengths. For panels b and c, markers show medians over eight different random networks of each length; error bars show interquartile range.

Grahic Jump Location
Fig. 6

Effect of dimensionless fiber bending stiffness κ. Results shown here are for a fiber length of L = 0.20 inclusion radii. (a) Displacement decay rate n for each bending stiffness κ. The increase in n with increasing κ indicates that as bending stiffness increases, the displacements propagate over a shorter range. (b) Nonaffinity χ̃ for each bending stiffness. For both panels, markers show medians of eight different random networks; error bars show interquartile range.

Grahic Jump Location
Fig. 7

Fiber reorientation. (a) and (b) Fiber reorientation, defined as the change in fiber angle, for the highest contraction with κ = 10−6 (a) and κ = 3 × 10−4 (b) the inner region within 15 inclusion radii is shown. (c) Histograms of fiber reorientation at the highest contraction, for fibers within 15 inclusion radii, averaged over the eight networks. As the fiber bending stiffness decreases, the fiber reorientation increases, but there is minimal change in reorientation below κ = 10−5. (inset) Histograms for fiber reorientation within a smaller inner region (<6 inclusion radii) shown by the solid circles in panels a and b. The same trends with respect to fiber bending stiffness hold closer to the inclusion where the displacements are larger. (d) Histograms for radial alignment (angle between each fiber and the radial direction) at the highest contraction, averaged over the eight networks. The radial alignment shows very little dependence on fiber bending stiffness.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In