Simulation of constant rate loading of eukaryotic cells using statistical thermodynamic methods

A. S. Nikityuk

Matematičeskaâ biologiâ i bioinformatika, Tome 19 (2024), pp. 304-321.

Résumé

The research is devoted to the study of the process of eukaryotic cell loading at a constant rate based on a statistical-thermodynamic model considering the orientational properties of the cytoskeleton. The presented results demonstrate the sufficiency of the considered model representation, as well as its potential for detailed study of the evolution of the filament network. It is assumed that the actin cytoskeleton contributes mainly to the mechanical response of the cell. In accordance with this hypothesis, an order parameter of filament segments characterizing their current orientation is introduced. Using self-consistent field methods, the free energy dependent on this parameter is obtained. This characteristic allows us to assess scenarios of the evolution of the actin cytoskeleton structure. Following the theory of linear thermodynamics, an evolutionary equation describing the mechanical behavior of a representative volume of eukaryotic cells that satisfy the basic thermodynamic laws is obtained. Analytical dependencies obtained using a parallel combination of Scott–Blair fractional elements are considered as test data against which the check of the examined model is performed to verify its suitability. This mechanical analog approximates quite accurately the results of atomic force microscopy measurements for a wide class of cells, but does not allow studying the process of microstructure evolution. The problem of optimization of parameters of the statistical-thermodynamic model of a cell in comparison with a fractional model has been formulated and solved. The results of solutions of the statistical-thermodynamic model with the selected parameters are in good qualitative and quantitative agreement with the test dependencies. Variation of the value of the relative calculation error from the value of the step of integration of the evolutionary differential equation of the representative volume of the cell is presented in order to confirm the reliability of the obtained results of modeling the process of loading with constant rate.

Bibliographie

[1] A. Po, A. Giuliani, M. G. Masiello, A. Cucina, A. Catizone, G. Ricci, M. Chiacchiarini, M. Tafani, E. Ferretti, M. Bizzarri, “Phenotypic transitions enacted by simulated microgravity do not alter coherence in gene transcription profile”, npj Microgravity, 5:1 (2019), 1–13 <ext-link ext-link-type='doi' href='https://doi.org/10.1038/s41526-018-0061-0'>10.1038/s41526-018-0061-0</ext-link><ext-link ext-link-type='mr-item-id' href='http://mathscinet.ams.org/mathscinet-getitem?mr=3908826'>3908826</ext-link>

[2] S. B. Han, J. K. Kim, G. Lee, D. H. Kim, “Mechanical Properties of Materials for Stem Cell Differentiation”, Adv. Biosyst, 4:11 (2020), 2000247 <ext-link ext-link-type='doi' href='https://doi.org/10.1002/adbi.202000247'>10.1002/adbi.202000247</ext-link>

[3] A. Isomursu, K. Y. Park, J. Hou, B. Cheng, M. Mathieu, G. A. Shamsan, B. Fuller, J. Kasim, M. M. Mahmoodi, T. J. Lu et al, “Directed cell migration towards softer environments”, Nat. Mater, 21:9 (2022), 1081–1090 <ext-link ext-link-type='doi' href='https://doi.org/10.1038/s41563-022-01294-2'>10.1038/s41563-022-01294-2</ext-link>

[4] A. Lesne, P. Bourgine, Morphogenesis: Origins of patterns and shapes, Springer Science & Business Media, 2011

[5] M. J. Paszek, N. Zahir, K. R. Johnson, J. N. Lakins, G. I. Rozenberg, A. Gefen, C. A. Reinhart King, S. S. Margulies, M. Dembo, D. Boettiger et al, “Tensional homeostasis and the malignant phenotype”, Cancer Cell, 8:3 (2005), 241–254 <ext-link ext-link-type='doi' href='https://doi.org/10.1016/j.ccr.2005.08.010'>10.1016/j.ccr.2005.08.010</ext-link><ext-link ext-link-type='mr-item-id' href='http://mathscinet.ams.org/mathscinet-getitem?mr=2126861'>2126861</ext-link>

[6] B. Zbiral, A. Weber, M. d. M. Vivanco, J. L. Toca-Herrera, “Characterization of Breast Cancer Aggressiveness by Cell Mechanics”, Int. J. Mol. Sci, 24:15 (2023), 12208 <ext-link ext-link-type='doi' href='https://doi.org/10.3390/ijms241512208'>10.3390/ijms241512208</ext-link>

[7] P. C. Georges, J. J. Hui, Z. Gombos, M. E. McCormick, A. Y. Wang, M. Uemura, R. Mick, P. A. Janmey, E. E. Furth, R. G. Wells, “Increased stiffness of the rat liver precedes matrix deposition: Implications for fibrosis”, Am. J. Physiol. Gastrointest. Liver Physiol., 293:6 (2007), G1147-G1154 <ext-link ext-link-type='doi' href='https://doi.org/10.1152/ajpgi.00032.2007'>10.1152/ajpgi.00032.2007</ext-link>

[8] G. Mattei, A. Ahluwalia, “Sample testing and analysis variables affecting liver mechanical properties: A review”, Acta Biomaterialia, 45 (2016), 60–71 <ext-link ext-link-type='doi' href='https://doi.org/10.1016/j.actbio.2016.08.055'>10.1016/j.actbio.2016.08.055</ext-link>

[9] G. S. van Tienderen, J. Conboy, I. Muntz, J. Willemse, J. Tieleman, K. Monfils, I. J. Schurink, J. A.A. Demmers, M. Doukas, G. H. Koenderink et al, “Tumor decellularization reveals proteomic and mechanical characteristics of the extracellular matrix of primary liver cancer”, Biomater. Adv, 146 (2023), 213289 <ext-link ext-link-type='doi' href='https://doi.org/10.1016/j.bioadv.2023.213289'>10.1016/j.bioadv.2023.213289</ext-link>

[10] A. Puliafito, L. Hufnagel, P. Neveu, S. Streichan, A. Sigal, D. K. Fygenson, B. I. Shraiman, “Collective and single cell behavior in epithelial contact inhibition”, Proc. Natl. Acad. Sci. U. S. A, 109:3 (2012), 739–744 <ext-link ext-link-type='doi' href='https://doi.org/10.1073/pnas.1007809109'>10.1073/pnas.1007809109</ext-link>

[11] B. Suki, J. H.T. Bates, “Lung tissue mechanics as an emergent phenomenon”, J. Appl. Physiol, 110:4 (2011), 1111–1118 <ext-link ext-link-type='doi' href='https://doi.org/10.1152/japplphysiol.01244.2010'>10.1152/japplphysiol.01244.2010</ext-link>

[12] E. Moeendarbary, A. R. Harris, “Cell mechanics: Principles, practices, and prospects”, Wiley Interdiscip. Rev. Syst. Biol. Med, 6:5 (2014), 371–388 <ext-link ext-link-type='doi' href='https://doi.org/10.1002/wsbm.1275'>10.1002/wsbm.1275</ext-link>

[13] J. S. de Sousa, R. S. Freire, F. D. Sousa, M. Radmacher, A. F.B. Silva, M. V. Ramos, A. C.O. Monteiro-Moreira, F. P. Mesquita, M. E.A. Moraes, R. C. Montenegro, C. L.N. Oliveira, “Double power-law viscoelastic relaxation of living cells encodes motility trends”, Sci. Rep, 10:1 (2020), 1–9 <ext-link ext-link-type='doi' href='https://doi.org/10.1038/s41598-019-56847-4'>10.1038/s41598-019-56847-4</ext-link><ext-link ext-link-type='mr-item-id' href='http://mathscinet.ams.org/mathscinet-getitem?mr=3948892'>3948892</ext-link>

[14] E. M. Darling, S. Zauscher, F. Guilak, “Viscoelastic properties of zonal articular chondrocytes measured by atomic force microscopy”, Osteoarthr. Cartil, 14:6 (2006), 571–579 <ext-link ext-link-type='doi' href='https://doi.org/10.1016/j.joca.2005.12.003'>10.1016/j.joca.2005.12.003</ext-link>

[15] W. J. Eldridge, S. Ceballos, T. Shah, H. S. Park, Z. A. Steelman, S. Zauscher, A. Wax, “Shear Modulus Measurement by Quantitative Phase Imaging and Correlation with Atomic Force Microscopy”, Biophys. J. Biophysical Society, 117:4 (2019), 696–705

[16] Y. M. Efremov, S. L. Kotova, P. S. Timashev, “Viscoelasticity in simple indentation-cycle experiments: a computational study”, Sci. Rep, 10:1 (2020), 1–15 <ext-link ext-link-type='doi' href='https://doi.org/10.1038/s41598-020-70361-y'>10.1038/s41598-020-70361-y</ext-link>

[17] P. Kollmannsberger, B. Fabry, “Linear and nonlinear rheology of living cells”, Annu. Rev. Mater. Res, 41 (2011), 75–97 <ext-link ext-link-type='doi' href='https://doi.org/10.1146/annurev-matsci-062910-100351'>10.1146/annurev-matsci-062910-100351</ext-link>

[18] X. Guo, G. Yan, L. Benyahia, S. Sahraoui, “Fitting stress relaxation experiments with fractional Zener model to predict high frequency moduli of polymeric acoustic foams”, Mech. Time-Dependent Mater, 20:4 (2016), 523–533 <ext-link ext-link-type='doi' href='https://doi.org/10.1007/s11043-016-9310-3'>10.1007/s11043-016-9310-3</ext-link>

[19] H. Babahosseini, B. Carmichael, J. S. Strobl, S. N. Mahmoodi, M. Agah, “Sub-cellular force microscopy in single normal and cancer cells”, Biochem. Biophys. Res. Commun, 463:4 (2015), 587–592 <ext-link ext-link-type='doi' href='https://doi.org/10.1016/j.bbrc.2015.05.100'>10.1016/j.bbrc.2015.05.100</ext-link>

[20] B. Carmichael, H. Babahosseini, S. N. Mahmoodi, M. Agah, “The fractional viscoelastic response of human breast tissue cells”, Phys. Biol, 12:4 (2015), 046001 <ext-link ext-link-type='doi' href='https://doi.org/10.1088/1478-3975/12/4/046001'>10.1088/1478-3975/12/4/046001</ext-link>

[21] V. M. Laurent, R. Fodil, P. Canadas, S. Fereol, B. Louis, E. Planus, D. Isabey, “Partitioning of cortical and deep cytoskeleton responses from transient magnetic bead twisting”, Ann. Biomed. Eng, 31:10 (2003), 1263–1278 <ext-link ext-link-type='doi' href='https://doi.org/10.1114/1.1616932'>10.1114/1.1616932</ext-link>

[22] S. Moreno-Flores, R. Benitez, M. D.M. Vivanco, J. L. Toca-Herrera, “Stress relaxation and creep on living cells with the atomic force microscope: A means to calculate elastic moduli and viscosities of cell components”, Nanotechnology, 21:44 (2010), 445101 <ext-link ext-link-type='doi' href='https://doi.org/10.1088/0957-4484/21/44/445101'>10.1088/0957-4484/21/44/445101</ext-link>

[23] Y. M. Efremov, T. Okajima, A. Raman, “Measuring viscoelasticity of soft biological samples using atomic force microscopy”, Soft Matter, 16:1 (2019), 64–81 <ext-link ext-link-type='doi' href='https://doi.org/10.1039/C9SM01020C'>10.1039/C9SM01020C</ext-link>

[24] A. R. Brazhe, N. A. Brazhe, N. N. Rodionova, A. I. Yusipovich, P. S. Ignatyev, G. V. Maksimov, E. Mosekilde, O. V. Sosnovtseva, “Non-invasive study of nerve fibres using laser interference microscopy”, Philos. Trans. R. Soc. A Math. Phys. Eng. Sci, 366:1880 (2008), 3463–3481 <ext-link ext-link-type='doi' href='https://doi.org/10.1098/rsta.2008.0107'>10.1098/rsta.2008.0107</ext-link>

[25] Y. Beloglazova, A. Nikitiuk, A. Voronina, O. Gagarskikh, Y. Bayandin, O. Naimark, V. Grishko, “Label-free single cell viability assay using laser interference microscopy”, Biology (Basel), 10:7 (2021)

[26] K. J. Van Vliet, G. Bao, S. Suresh, “The biomechanics toolbox: Experimental approaches for living cells and biomolecules”, Acta Mater, 51:19 (2003), 5881–5905 <ext-link ext-link-type='doi' href='https://doi.org/10.1016/j.actamat.2003.09.001'>10.1016/j.actamat.2003.09.001</ext-link>

[27] Y. Hao, S. Cheng, Y. Tanaka, Y. Hosokawa, Y. Yalikun, M. Li, “Mechanical properties of single cells: Measurement methods and applications”, Biotechnol. Adv, 45 (2020), 107648 <ext-link ext-link-type='doi' href='https://doi.org/10.1016/j.biotechadv.2020.107648'>10.1016/j.biotechadv.2020.107648</ext-link>

[28] M. V. Shitikova, “Fractional Operator Viscoelastic Models in Dynamic Problems of Mechanics of Solids: A Review”, Mech. Solids, 57:1 (2022), 1–33 <ext-link ext-link-type='doi' href='https://doi.org/10.3103/S0025654422010022'>10.3103/S0025654422010022</ext-link><ext-link ext-link-type='zbl-item-id' href='https://zbmath.org/?q=an:1511.74009'>1511.74009</ext-link>

[29] A. S. Nikitiuk, “Parameter identification of viscoelastic cell models based on force curves and wavelet transform”, Comput. Res. Model, 15:6 (2023), 1653–1672 <ext-link ext-link-type='doi' href='https://doi.org/10.20537/2076-7633-2023-15-6-1653-1672'>10.20537/2076-7633-2023-15-6-1653-1672</ext-link>

[30] A. S. Nikitiuk, A. A. Koshkina, Y. V. Bayandin, O. B. Naimark, “On thermodynamics and relaxation properties of eukaryotic cells”, Int. J. Non. Linear. Mech, 157 (2023), 104532 <ext-link ext-link-type='doi' href='https://doi.org/10.1016/j.ijnonlinmec.2023.104532'>10.1016/j.ijnonlinmec.2023.104532</ext-link>

[31] H. Lu, B. Wang, J. Ma, G. Huang, H. Viswanathan, “Measurement of creep compliance of solid polymers by nanoindentation”, Mech. Time-Dependent Mater., 7:3-4 (2003), 189–207

[32] A. P. Kren, A. S. Machikhin, M. F. Bulatov, “Determination of the creep function using atomic force microscope”, Mater. Lett, 259 (2020), 126872 <ext-link ext-link-type='doi' href='https://doi.org/10.1016/j.matlet.2019.126872'>10.1016/j.matlet.2019.126872</ext-link>

[33] F. Gittes, B. Schnurr, P. D. Olmsted, F. C. MacKintosh, C. F. Schmidt, “Microscopic viscoelasticity: Shear moduli of soft materials determined from thermal fluctuations”, Phys. Rev. Lett, 79:17 (1997), 3286 <ext-link ext-link-type='doi' href='https://doi.org/10.1103/PhysRevLett.79.3286'>10.1103/PhysRevLett.79.3286</ext-link>

[34] F. Ziemann, J. Radler, E. Sackmann, “Local measurements of viscoelastic moduli of entangled actin networks using an oscillating magnetic bead micro-rheometer”, Biophys. J., 66:6 (1994), 2210–2216 <ext-link ext-link-type='doi' href='https://doi.org/10.1016/S0006-3495(94)81017-3'>10.1016/S0006-3495(94)81017-3</ext-link>

[35] F. Mainardi, Spada G. Creep, “relaxation and viscosity properties for basic fractional models in rheology”, Eur. Phys. J. Spec. Top, 193:1 (2011), 133–160 <ext-link ext-link-type='doi' href='https://doi.org/10.1140/epjst/e2011-01387-1'>10.1140/epjst/e2011-01387-1</ext-link>

[36] H. Xu, X. Jiang, “Creep constitutive models for viscoelastic materials based on fractional derivatives”, Comput. Math. with Appl, 73:6 (2017), 1377–1384 <ext-link ext-link-type='doi' href='https://doi.org/10.1016/j.camwa.2016.05.002'>10.1016/j.camwa.2016.05.002</ext-link><ext-link ext-link-type='mr-item-id' href='http://mathscinet.ams.org/mathscinet-getitem?mr=3623128'>3623128</ext-link><ext-link ext-link-type='zbl-item-id' href='https://zbmath.org/?q=an:1458.74027'>1458.74027</ext-link>

[37] Y. Gao, B. Zhao, D. Yin, L. Yuan, “A general fractional model of creep response for polymer materials: Simulation and model comparison”, J. Appl. Polym. Sci, 139:5 (2022) <ext-link ext-link-type='zbl-item-id' href='https://zbmath.org/?q=an:1521.70004'>1521.70004</ext-link>

[38] T. Demir, Mittag-Leffler Function and Integration of the Mittag-Leffler Function, March 2022

[39] D. C. Lin, D. I. Shreiber, E. K. Dimitriadis, F. Horkay, “Spherical indentation of soft matter beyond the Hertzian regime: Numerical and experimental validation of hyperelastic models”, Biomech. Model. Mechanobiol, 8:5 (2009), 345–358 <ext-link ext-link-type='doi' href='https://doi.org/10.1007/s10237-008-0139-9'>10.1007/s10237-008-0139-9</ext-link>

[40] A. S. Nikitiuk, E. A. Korznikova, S. V. Dmitriev, O. B. Naimark, “Dna breathers and cell dynamics”, Math. Biol. Bioinforma, 14:1 (2019) <ext-link ext-link-type='doi' href='https://doi.org/10.17537/2019.14.137'>10.17537/2019.14.137</ext-link>

[41] J. B. Spinelli, E. Zaganjor, “Mitochondrial efficiency directs cell fate”, Nat. Cell Biol, 24:2 (2022), 125–126 <ext-link ext-link-type='doi' href='https://doi.org/10.1038/s41556-021-00834-3'>10.1038/s41556-021-00834-3</ext-link>

[42] K. Wang, Y. Qin, Y. Chen, “In situ AFM detection of the stiffness of the in situ exposed cell nucleus”, Biochim. Biophys. Acta Mol. Cell Res, 1868:5 (2021), 118985 <ext-link ext-link-type='doi' href='https://doi.org/10.1016/j.bbamcr.2021.118985'>10.1016/j.bbamcr.2021.118985</ext-link>

[43] W. Yu, S. Sharma, E. Rao, A. C. Rowat, J. K. Gimzewski, D. Han, J. Rao, “Cancer cell mechanobiology: a new frontier for cancer research”, J. Natl. Cancer Cent, 2:1 (2022), 10–17 <ext-link ext-link-type='doi' href='https://doi.org/10.1016/j.jncc.2021.11.007'>10.1016/j.jncc.2021.11.007</ext-link>

[44] P. G. de Gennes, J. Prost, R. Pelcovits, “The Physics of Liquid Crystals”, Phys. Today, 48:5 (1995), 70–71 <ext-link ext-link-type='doi' href='https://doi.org/10.1063/1.2808028'>10.1063/1.2808028</ext-link>

[45] M. I. Shliomos, Y. L. Raikher, “Orientational ordering and mechanical properties of solid polymers”, J. Exp. Theor. Phys, 5 (1978), 1760–1783

[46] V. S. Deshpande, R. M. McMeeking, A. G. Evans, “A bio-chemo-mechanical model for cell contractility”, Proc. Natl. Acad. Sci. U. S. A, 103:38 (2006), 14015–14020 <ext-link ext-link-type='doi' href='https://doi.org/10.1073/pnas.0605837103'>10.1073/pnas.0605837103</ext-link>

[47] A. S. Nikitiuk, Y. V. Bayandin, O. B. Naimark, “Statistical thermodynamics of DNA with open states”, Phys. A Stat. Mech. its Appl, 607 (2022), 128156 <ext-link ext-link-type='doi' href='https://doi.org/10.1016/j.physa.2022.128156'>10.1016/j.physa.2022.128156</ext-link><ext-link ext-link-type='mr-item-id' href='http://mathscinet.ams.org/mathscinet-getitem?mr=4484872'>4484872</ext-link><ext-link ext-link-type='zbl-item-id' href='https://zbmath.org/?q=an:07614907'>07614907</ext-link>

[48] D. A. Bilalov, Y. V. Bayandin, O. B. Naimark, “Mathematical modeling of failure process of AlMg2.5 alloy during highand very high cycle fatigue”, Comput. Contin. Mech, 11:3 (2018), 323–334 <ext-link ext-link-type='doi' href='https://doi.org/10.7242/1999-6691/2018.11.3.24'>10.7242/1999-6691/2018.11.3.24</ext-link>

[49] L. M. Rebelo, J. S. De Sousa, J. Mendes Filho, M. Radmacher, “Comparison of the viscoelastic properties of cells from different kidney cancer phenotypes measured with atomic force microscopy”, Nanotechnology, 24:5 (2013), 055102 <ext-link ext-link-type='doi' href='https://doi.org/10.1088/0957-4484/24/5/055102'>10.1088/0957-4484/24/5/055102</ext-link>

[50] L. M. Rebelo, J. S. De Sousa, J. M. Filho, J. Schape, H. Doschke, M. Radmacher, “Microrheology of cells with magnetic force modulation atomic force microscopy”, Soft Matter, 10:13 (2014), 2141–2149 <ext-link ext-link-type='doi' href='https://doi.org/10.1039/C3SM52045E'>10.1039/C3SM52045E</ext-link>

[51] A. Rigato, A. Miyagi, S. Scheuring, F. Rico, “High-frequency microrheology reveals cytoskeleton dynamics in living cells”, Nat. Phys, 13:8 (2017) <ext-link ext-link-type='doi' href='https://doi.org/10.1038/nphys4104'>10.1038/nphys4104</ext-link>

[52] A. R. Bausch, F. Ziemann, A. A. Boulbitch, K. Jacobson, E. Sackmann, “Local measurements of viscoelastic parameters of adherent cell surfaces by magnetic bead microrheometry”, Biophys. J., 75:4 (1998), 2038–2049 <ext-link ext-link-type='doi' href='https://doi.org/10.1016/S0006-3495(98)77646-5'>10.1016/S0006-3495(98)77646-5</ext-link>

[53] Y. M. Efremov, W. H. Wang, S. D. Hardy, R. L. Geahlen, A. Raman, “Measuring nanoscale viscoelastic parameters of cells directly from AFM force-displacement curves”, Sci. Rep, 7:1 (2017), 1–14 <ext-link ext-link-type='doi' href='https://doi.org/10.1038/s41598-017-01784-3'>10.1038/s41598-017-01784-3</ext-link>

[54] O. B. Naimark, Y. V. Bayandin, M. A. Zocher, “Collective properties of defects, multiscale plasticity, and shock induced phenomena in solids”, Phys. Mesomech, 20:1 (2017), 10–30 <ext-link ext-link-type='doi' href='https://doi.org/10.1134/S1029959917010027'>10.1134/S1029959917010027</ext-link>

[55] Roylance D., “Engineering viscoelasticity”, Modules in Mechanics of Materials, Cambridge Massachusetts Institute of Technology, 2001 (data obrascheniya: 20.04.2024) <ext-link ext-link-type='uri' href='https://web.mit.edu/course/3/3.11/www/modules/visco.pdf'>https://web.mit.edu/course/3/3.11/www/modules/visco.pdf</ext-link>

[56] O. Thoumine, A. Ott, “Time scale dependent viscoelastic and contractile regimes in fibroblasts probed by microplate manipulation”, J. Cell Sci, 110:17 (1997), 2109–2116 <ext-link ext-link-type='doi' href='https://doi.org/10.1242/jcs.110.17.2109'>10.1242/jcs.110.17.2109</ext-link>

[57] B. Laperrousaz, G. Drillon, L. Berguiga, F. Nicolini, B. Audit, V. M. Satta, A. Arneodo, F. Argoul, “From elasticity to inelasticity in cancer cell mechanics: A loss of scale invariance”, AIP Conf Proc., 1760:1 (2016)