Abstract
We develop equilibrium fluctuation formulae for the isothermal elastic moduli of discrete biomembrane models at different scales. We account for the coupling of large stretching and bending strains of triangulated network models endowed with harmonic and dihedral angle potentials, on the basis of the discrete-continuum approach presented in Schmidt and Fraternali (J Mech Phys Solids 60:172–180, 2012). We test the proposed equilibrium fluctuation formulae with reference to a coarse-grained molecular dynamics model of the red blood cell (RBC) membrane (Marcelli et al. in Biophys J 89:2473–2480, 2005; Hale et al. in Soft Matter 5:3603–3606, 2009), employing a local maximum-entropy regularization of the fluctuating configurations (Fraternali et al. in J Comput Phys 231:528–540, 2012). We obtain information about membrane stiffening/softening due to stretching, curvature, and microscopic undulations of the RBC model. We detect local dependence of the elastic moduli over the RBC membrane, establishing comparisons between the present theory and different approaches available in the literature.
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References
Arrojo M, Ortiz M (2006) Local maximum-entropy approximation schemes: a seamless bridge between finite elements and meshfree methods. Int J Numer Methods Eng 65: 2167–2202
Borelli MES, Kleinert H, Schakel AMJ (1999) Derivative expansion of one-loop effective energy of stiff membranes with tension. Phys Lett A 253: 239–246
Cyron CJ, Arrojo M, Ortiz M (2009) Smooth, second-order, non-negative meshfree approximants selected by maximum entropy. Int J Numer Methods Eng 79: 1605–1632
Dao M, Li J, Suresh S (2006) Molecularly based analysis of deformation of spectrin network and human erythrocyte. Mater Sci Eng 26: 1232–1244
Discher DE, Boal DH, Boey SK (1997) Phase transitions and anisotropic responses of planar triangular nets under large deformation. Phys Rev E 55(4): 4762–4772
Fedosov DA, Caswell B, Karniadakis GE (2009) General coarse-grained red blood cell models: I. mechanics. ArXiv e-prints
Fraternali F, Lorenz C, Marcelli G (2012) On the estimation of the curvatures and bending rigidity of membrane networks via a local maximum-entropy approach. J Comput Phys 231: 528–540
Gibson L, Ashby M (1982) The mechanics of three-dimensional cellular materials. Proc R Soc Lond A Mat 382(1782): 43–59
Gompper G, Kroll D (1996) Random surface discretization and the renormalization of the bending rigidity. J Phys I Fr 6: 1305–1320
Hale J, Marcelli G, Parker K, Winlowe C, Petrov G (2009) Red blood cell thermal fluctuations: comparison between experiment and molecular dynamics simulations. Soft Matter 5: 3603–3606
Hartmann D (2010) A multiscale model for red blood cell mechanics. Biomech Model Mechanobiol 9: 1–17
Helfrich W (1985) Effect of thermal undulations on the rigidity of fluid membranes and interfaces. J Phys 46: 1263–1268
Helfrich W (1998) Stiffening of fluid membranes and entropy loss of membrane closure: two effects of thermal undulations. Eur Phys J B 1: 481–489
Helfrich W, Kozlov MM (1993) Bending tensions and the bending rigidity of fluid membranes. J Phys II Fr 3: 287–292
Helfrich W, Servuss R (1984) Untlulations, steric interactian and cohesion of fldd untlulations, steric interaction and cohesion of fluid membranes. Nuovo Cimento 1: 137–151
Hess S, Kröger M, Hoover W (1997) Shear modulus of fluids and solids. Phys A 239
Holzapfel GA (2000) Nonlinear solid mechanics, a continuum approach for engineering. Wiley, Chichester
Kleinert H (1986) Thermal softening of curvature elasticity in membranes. Phys Lett A 114: 263–268
Kohyama T (2009) Simulations of flexible membranes uding a coarse-grained particle-based model with spontaneous curvature variables. Phys A 388: 3334–3344
Kühnel W (2002) Differential geometry, curves-surfaces-manifolds. American Mathematical Society, Providence, RI
Lee J, Discher D (2001) Deformation-enhanced fluctuations in the red cell skeleton with theoretical relations to elasticity, connectivity, and spectrin unfolding. Biophys J 81: 3178–3192
Lipowsky R, Girardet M (1990) Shape fluctuations of polymerized or solidlike membranes. Phys Rev Lett 65(23): 2893–2896
Lutsko J (1989) Generalized expressions for the calculation of the elastic constants by computer simulation. J Appl Phys 8: 2991–2997
Marcelli G, Parker H, Winlove P (2005) Thermal fluctuations of red blood cell membrane via a constant-area particle-dynamics model. Biophys J 89: 2473–2480
Mecke KR (1995) Bending rigidity of fluctuating membranes. Z Phys B Condens Mater 97: 379–387
Müller M, Katsov K, Schick M (2006) Biological and syntetic membranes: what can be learned from a coarse-grained description?. Phys Rep 434: 113–176
Munkres J (1984) Elements of algebraic topology. Addison-Wesley, Menlo Park, CA
Naghdi PM (1972) The theory of shells and plates. In: Flügge’s S (eds) Handbuch der Physik, Vol. VIa/2, C. Trusdell Ed.. Springer, Berlin, pp 425–640
Nelson D, Piran T, Weinberg S, (eds) (2004) Statistical mechanics of membranes and surfaces, 2nd edn. World Scientific, Singapore
Ogden RW (1984) Non-linear elastic deformations. Dover, Mineola
Onck P, Koeman T, van Dillen T, van der Glessen E (2005) Alternative explanation of stiffening in cross-linked semiflexible networks. Phys Rev Lett 95
Peliti L, Leibler S (1985) Effects of thermal fluctuations on systems with small surface tensionl fluctuations on systems with small surface tension. Phys Rev Lett 54: 1690–1693
Pinnow H, Helfrich W (2000) Effect of thermal undulations on the bending elasticity and spontaneous curvature of fluid membranes. Eur Phys J E 3: 149–157
Pivkin I, Karniadakis G (2008) Accurate coarse-grained modeling of red blood cells. Phys Rev Lett 101
Ray J, Rahman A (1984) Statistical ensembles and molecular dynamics studies of anisotropic solids. J Chem Phys 80(9): 4423–4428
Schmidt B (2006) A derivation of continuum nonlinear plate theory from atomistic models. SIAM Multiscale Model Simul 5: 664–694
Schmidt B (2008) On the passage from atomic to continuum theory for thin films. Arch Ration Mech Anal 190: 1–55
Schmidt B, Fraternali F (2012) Universal formulae for the limiting elastic energy of static membrane networks. J Mech Phys Solids 60: 172–180
Schöffel P, Möser MH (2001) Elastic constants of quantum solids by path integral simulations. Phys Rev B 63(224108): 1–9
Seung H, Nelson D (1988) Defects in flexible membranes with crystalline order. Phys Rev A 38: 1005–1018
Smith W, Forester T (1999) The dl_poly_2 molecular simulation package. http://www.cse.clrc.ac.uk/msi/software/DL_POLY
Squire D, Holt A, Hoover W (1969) Isothermal elastic constants for argon. Theory and monte carlo calculations. Physica 42: 388–397
Tu ZC, Ou-Yang ZC (2008) Elastic theory of low-dimensionale continua and its application in bio- and nano-structures. J Comput Theor Nanosci 5: 422–448
Yoshimoto K, Papakonstantopoulos G, Lutsko J, de Pablo J (2005) Statistical calculation of the elastic moduli for atomistic models. Phys Rev B 71(181108): 1–6
Zhou Z, Joós B (1996) Stability criteria for homogeneously stressed materials and the calculation of elastic constants. Phys Rev B 54(6): 3841–3850
Zhou Z, Joós B (1997) Mechanisms of membrane rupture: from cracks to pores. Phys Rev B 56: 2997–3009
Zhou Z, Joós B (1999) Convergence issues in molecular dynamics simulations of highly entropic materials. Model Simul Mater Sci Eng 7: 383–395
Zhou Z, Joós B (2002) Fluctuation formulas for the elastic constants of an arbitrary system. Phys Rev B 66
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Fraternali, F., Marcelli, G. A multiscale approach to the elastic moduli of biomembrane networks. Biomech Model Mechanobiol 11, 1097–1108 (2012). https://doi.org/10.1007/s10237-012-0376-9
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DOI: https://doi.org/10.1007/s10237-012-0376-9