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Journal Club September 2010: Modeling the Mechanics of Cellular Membranes

Alexander A. Spector's picture

Constitutive relations, 2-D vs. 3-D. The starting point for modeling cellular membranes is the constitutive relations in 2-D space. It is important to set up the corresponding equations directly in two dimensions rather than to consider them as an asymptotic limit of three-dimensional relationships, like it is done in the shell theory. The main reason for the direct 2-D relations is that 3-D continuum approaches are not applicable to membranes whose thickness in on the order of magnitude of the dimension of a single molecule.

Solid-like membranes. Red blood cell membrane. Some cellular membranes (e.g., the membranes of the red blood cell or cochlear outer hair cell), where the plasma membrane is attached to the underlying 2-D cytoskeleton, can be treated as elastic or viscoelastic solids. The model of the red blood cell membrane can be considered as a gold standard in the analysis of cellular membranes, and it is based on nonlinear viscoelastic relationships, including the bending mode. Nonlinearity is critical for the description of the large deformations during cell's squeezing through narrow capillaries, and the bending component is important for the modeling of cell shape inside the blood flow. The red blood cell constitutive relations and their computational implementations developed earlier have broad applications today. One important application (for review, Pozrikidis [1]) is the simulation of the cell movement through the blood flow, including the interactions among red blood cells in the rouleau. Another application is the modeling of red blood cells in disease. As an important example, it has been shown that the mechanical properties change significantly in malaria. Suresh et al. [2] demonstrated how the red blood cell shear modulus, the main parameter of the cell's membrane mechanics, increases with different stages of the P. falciparum parasite maturation in the cell.

Liquid-like membranes. Membrane tethers. Pure plasma membranes have to be considered as special fluids whose primary properties are tension and bending. From the standpoint of general continuum mechanics, it has been shown that the strain energy functional (function) of liquid membranes is determined by the curvatures and local area change. In many cases, liquid membranes are considered as completely unstretchable, and the area preservation condition is introduced via a Lagrange multiplier. In 1973, Herfrich proposed a broadly used energy functional in the form of a function of the membrane curvature after deformation and the membrane original (spontaneous) curvature. In axisymmetric cases, the Euler-Lagrange equation for this functional can be explicitly derived in the form of a nonlinear 4-th order ODE and solved by using asymptotic and finite difference methods (e.g., Derenyi et al. [3]). Alternatively, finite element methods can be directly applied to the minimization of the bending energy functional. Membrane tethers (narrow tubes) are an interesting and general phenomenon in liquid membranes. Such tethers can form naturally for cell-cell communication or for the slowing down of moving cells (such as leukocytes). Pulling membrane tethers is also an effective method to probe the membrane local properties. There has been a significant effort to model this membrane phenomenon. An approach based on the thermodynamic balance of a tether system was previously applied to estimate the membrane bending modulus, membrane-cytoskeleton adhesion energy, and tension. The details of shape of a narrow tether are often unavailable in light microscopy. One approach to resolve this problem is to combine the tether pulling experiment with micropipette aspiration of the membrane on the opposite side of the cell and to compute the radius of the tether based on the length of the aspirated part of the membrane inside the pipette. This approach was successfully applied to the estimation of the membrane-cytoskeleton adhesion and membrane tension in red blood cells. Alternatively, membrane shape in the whole tether region can be computed from the analysis of the equilibrium state or steady state deformation of the tether (Derenyi et al. [3]). Such an approach was recently extended to model tether shape in different cells, taking into account particular arrangement of bonds between the membrane and the cellular cytoskeleton.

Electromechanical coupling in membranes.  Electromechanical coupling occurs naturally in biological membranes because they possess and are surrounded by a system of electric charges. One mode of such coupling bi-directionally relates the membrane curvature to an applied electric field. Physical models of this membrane phenomenon, called flexoelectricity, relate coupling between the membrane curvature and the electric field to a redistribution of the electrical charges associated with the bilayer (Harland el al. [4]). Another mode of electromechanical coupling in cellular membranes is associated with the recently discovered membrane protein prestin. The membranes (cells) containing this protein exhibit dimensional changes in response to the application of a transmembrane electric field, a phenomenon named electromotility. Similar to piezoelectric materials, these membranes generate (transfer) electric charges in response to membrane (cell) deformation. Originally, prestin was found in the cochlear outer hair cell, and it was shown that this protein is critical to the amplification and frequency selectivity of the mammalian ear. Lately, a number of originally nonmotile cells, such as human embryonic kidney cells, were transfected with prestin in order to use genetic manipulations in studies of electromotility. The membranes with prestin can be modeled by using mechanically linear (small deformations) and electrically nonlinear (saturating electric charge) piezoelectric-like constitutive relations. Prestin is probably the fastest known protein, and one of the main challenges for models of membranes with prestin is their applicability to high-frequency regimes (up to tens of kHz) of cell performance (for review, Spector et al. [5]).

References

[1] Pozrikidis, C. (Ed) Computational Hydrodynamics of Capsules and Biological Cells. Taylor&Francis, 2010.

[2] Suresh, S., Spatz, J., Mills, A., Dao, M., Lim, C.T., Beil, M., Seufferlein, T. Connections between single-cell biomechanics and human disease states: gastrointestinal cancer and malaria, Acta Biomater., 1, 15, 2005.

[3] Derenyi, I., Julicher, F., and Prost, J. Formation and interaction of membrane tubes, Phys. Rev. Letts, 88, 238101, 2002.

[4] Harland, B., Brownell, W.E., Spector, A.A., and Sun, S.X. Voltage-induced bending and electromechanical coupling in lipid bilayers, Phys. Rev. E., 81, Art. 031907, 2010.

[5] Spector, A.A., Deo, N., Grosh, K., Ratnanather, J.T., and Raphael, R.M. Electromechanical models of the outer hair cell composite membrane, J. Membr. Biol., 209, 135, 2006. 

 

 

Comments

Xiaodong Li's picture

Thank you for hosting this theme. It seems like there are some critical parameters that are needed for modeling. I would like to know what parameters (mechanical properties, for example) should be obtained from experiments. It may help if we can map local micro strains across the membranes. I am thinking about using digital image correlation to get this. Your comments and suggestions are appreciated. 

Alexander A. Spector's picture

 Dear Xiaodong,

 

Thank you for your interest in this topic and stimulating questions. In terms of solid-like elastic isotropic membranes, their main mechanical parameters are shear modulus, areal modulus, and bending modulus. Note that these parameters are 2-D versions of the conventional 3-D moduli and have units of force/length, force/length, and force x length, respectively.  In terms of liquid-like membranes, their main mechanical parameters are the bending modulus and tension. In the case of cellular membranes, the density of the adhesion energy is a critical parameter of the interaction between the plasma membrane and underlying cytoskeleton. There are a number of common techniques to extract these parameters, including micropipette aspiration and tether pulling. In terms of the measurement of membrane strains, I would like to note that the membrane displacements could be very small (on the order of a nm). As an example, I would like to mention our recent paper by Schumacher et al. (Phys. Rev. E., 80, 041905, 2009) where we computed the displacements and strains along cellular membranes. Hopefully, my response shed some additional light.

Hi Alexander, nice theme, relevant topic. I am interested in the mechanics of bacterial membranes. Do you know any text-book or review article on this? I am especially interested in the constitutive relations as a function of the underlying structure.

Cheers, Patrick

Alexander A. Spector's picture

Partick,

Thank you for your stimulating question. AFM seems is a common technique to study the mechanical properties of bacterial membranes. In this regard, I would mention two represntative papers: V. Vadillo-Rodriguez et al., Surface Viscoelasticity of Individual Gram-Negative Bacterial Cells Measured Using Atomic Force Miroscopy , J. Bacteriology, 190 (12),2008 and M. Arnoldi et al. Bacterial Turgor Pressure Can Be Measured by Atomic Force Microscopy, Phys. Rev. E, 62 (1), 2000. The first paper is based on the model of Standard Linear Solid for the interpretation of its experiment. The second paper uses the bending-associated free energy that I discussed in the original comment (see the section Liquid-Like Membranes).

Best, Alex.  

 

 

peppezurlo's picture

Dear Professor Spector, 

I am very happy you started this new topic since I am very interested in the modelling lipid bilayer thin membranes, which I studied during my PhD. I would like to point your attention on our work on a coherent deduction of bending/non local moduli from very simple measurements of the in-plane chemo-mechanical behavior of the lipid bilayer: Derivation of a new free energy for biological membranes, Cont. Mech. and Thermodyn. 20(5) 2008 255-273. We believe that the determination of a consistent energy density for such membranes can be afforded by a sort of asymptotic expansion of a parent 3D energy with respect to the thickness, with considerations that span from symmetry requirements which describe the liquid-like behavior, to assumptions on the single lipid kinematics.

The results are quite encouraging since these establish a precise relation among bending (mean and Gaussian) moduli, the surface tension and the chemical composition. 

Anyway I hope to see some update on this topic very soon!

All the best,

Giuseppe Zurlo

Alexander A. Spector's picture

Dear Giuseppe,

Thank you for keeping our discussion active and for pointing out at your and your colleagues' interesting paper on the free energy for biological membranes. The paper raises a fundamental question on a justification of the asymptotic 3-D approach to the derivation of membrane models. Another important point is how to treat the low stretchablity of membranes: is it 3-D volume preservation or 2-D area preservation? Furthermore, if it is the latter case what is the most effective way to provide it in computational models? These questions were discussed in the original post that started the discussion in September but I use this occasion to re-iterate them in connection to this interesting paper.

It is impressive that you were able to derive the model moduli in terms of the interaction of the membrane components. In this regard, it is of special interest to learn how to model the effect of cholesterol on the bilayer as well as on membrane proteins. This is obviously important from the medical standpoint, and numerous groups have collected the data on the effect of cholesterol concentration on the membrane/membrane protein properties. As an example, I would like to mention the paper Tuning of the Outer hair Cell Motor by Membrane Cholesterol by Rajagopalan et al. (J. Biol. Chemistry, 282, 2007, 36659-36670).

Best, Alex. 

 

peppezurlo's picture

Dear Professor Spector,

you raise an interesting point on the question: are lipid bilayers incompressible thin shells or are these area-preserving surfaces? This point plays the role of an ubiquitous faq in the modelization of fluid membranes.

Our standpoint in the modelization of GUVs is that the lipid membrane must not be considered as a fluid surface in the true sense of the word, rather as a very thin object whose thickness is typically up-to hundreds of times smaller that the GUVs' diameter.This point of view suggests that the membrane energy may be expanded with respect to the thickness, and that each factor of the different thickness powers plays a precise mechanical role. Our idea is that factors of the lower thickness powers play a relevant role in the overall membrane response, and that factors of the higher thickness powers play a sort of "refinement" role. 

Concerning the membrane compressibilty, as it is documented by a vaste experimental literature, the volume compressibility of lipid membranes is not zero, but actually it is so small so that it can be neglected during a wide variety of membrane deformations, and even during the drastic order-disorder transition (e.g. Goldstein & Leibler, Phys.Rev. A 40(2) 1989) .

Since the molecular volume is fairly constant, the product of the head-group area and the bilayer thickness should remain constant during the membrane deformations, which means that the lipid membrane cannot be considered - in general - as an area preserving surface. If the membrane were area preserving, the membrane thickness should remain constant too, leading to the impossibility of describing (for example) the membrane thinning and, furthermore, its transition to thicker (ordered) states from thinner (disordered) states with increasing temperature. Further, there is no need in postulating its in-plane-rigidity: the surface tension is elastically determined from the osmotic pressure: we have done calculations in the ordered and disordered states of what we call "non-local" moduli from informations regarding chemical composition. It would be interesting to discuss these results in light of experimental measurements. Up to your knowledge, are there collected and detailed data on the membrane stretchability and bending moduli for diffent compositions? Thank you for the article you pointed to our attention, which may be really useful in this direction! I will read it carefully!!! 

This is a great great blog! Thanks for posting it! :D

Cordially yours,

Giuseppe Zurlo

 

 

Alexander A. Spector's picture

Dear Giuseppe,

Thank you for your continuing interest in this topic.

Obviously, real membranes are 3-D objects. However, they are very thin, and  the main question is how to reduce the constitutive relations to effective 2-D ones. You are talking about an approach that is based on a power expansion of the membrane free energy with respect to the membrane dimensionless thickness. In such a case, there are two questions to be resolved. First, plasma membranes (lipid bilayers) are two-molecule thick, and a contnuum description might be a problem in this case. Second, even if we try to present the free energy as a continuous function of the tmembrane thickness, it might be a very heterogynous function to reflect various typical regions (phospholipid heads, tails, etc) as well as asymmetry of two bilayers, etc. Thus, its expansion might be quite nontrivial.

Another point that you discuss in your comment is the volume incompressibility vs the area incompressibility. The level of the membrane area compressibility (stretchability) is governed by the interaction of lipid molecules. Plasma membranes are normally low stretchable: they can sustain up to 2-3% of the local area change before they rupture. Several approaches have been used to incorporate this membrane feature into 2-D consitutive relations: 1) membrane is slightly stretchable and it is described by the corresponding term in the free energy that is quadratic with repect to the local area change; it is characterized by the area modulus, 2) membrane area change is characterized by typical tension, and it enters the free energy as a term linear with respect to the membrane area, and 3) the membrane is assumed to be fully untretchable, this condition is considered as a constraint in the energy minimization, and the corresponding tension appears as a Lagrange multiplyer. These are just the most common assumptions.

Finally, you raised a very interesting and stimulating question on how the membrane moduli depend on the membrane composition. In terms of the bending modulus, its ranges is between 10kT and 100kT. A relevant reference on this is the paper Coalescence of Membrane Tethers: Experiments, Theory, and Applications by D. Cuvelier et al. (Biophys. J., 88, 2005, 2714-2726). Another (not-unlerated) question is how the membrane moduli vary from cell to cell. We reviewed some of the data in the papers Modeling the Mechanics of Tethers Pulled From the Cochlear Outer Hair Cell by K. Schumacher et al. (Journ. of Biomechanical Eng., 130, 2008, Art. 031007) and Computational Analysis of the Tether-Pulling Experiment to Probe Plasma Membrane-Cytoskeleton Interaction in Cells by K. Schumacher et al. (Physical Review E, 80, 2009, Art. 041905).

Best regards,

Alex.

 

 

 

 

 

 

 

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