Skip to main content
SpringerLink
Log in

Molecular Mechanics of the Cell

  • Chapter
  • First Online:
The Physics of Living Systems

Part of the book series: Undergraduate Lecture Notes in Physics ((ULNP))

  • 2977 Accesses

Abstract

The mathematical models of polymers and membranes are introduced, to describe the microscopic deformation of biological materials. The mechanical properties at the molecular scale are inherently statistical, and remain hidden below macroscopic averages. But we can now see and manipulate a single molecule, and follow its movements in real time. This is made possible thanks to the advent of revolutionary tools, such as the atomic-force microscope and the optical or magnetic tweezers, which allow to test the mechanics of biological materials with unprecedented quality and accuracy, down to the molecular scale. Today it is possible to measure the elasticity of a single molecule, of a piece of DNA, of a fragment of cell membrane, and from such strictly physical comparisons, a totally new wealth of information has started to invade the already flooded desk of the biologist.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 64.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    For the definitions of the stress/strain tensors (\(\sigma , \varepsilon \)) and elastic moduli (\(E,\nu \)), see Appendix H on p. 422.

  2. 2.

    As a curiosity, it may be noted that Helfrich was at that time working with Hoffmann-LaRoche Laboratories in Switzerland, on the fabrication of the first liquid-crystals displays, a subject he had started years before when working at RCA in Princeton and for which he is a credited inventor.

  3. 3.

    It may be worth pointing out that the perturbation amplitude is described by a parameter \(\alpha =M_B(c_1q^4+c_2q^2+c_3)\), closely resembling the analysis of temperature fluctuations of the lipid membrane in Eqs. (8.61) through (8.65).

  4. 4.

    For the many details of this extraordinary process, the reader can consult any good biology textbook, such as the classical Molecular Biology of the Cell, by Alberts, Bray, Lewis, Raff, Roberts and Watson (Garland Science Pub., New York, 4th ed. 2002).

  5. 5.

    Notably, plant cells operate the membrane division by an entirely different mechanism. After chromosome splitting, a portion of the rigid cell wall that surrounds most vegetable cells is nucleated in a new plaque, around the residual microtubules from the spindle that form a temporary structure called phragmoplast. This plaque starts with a roughly discoidal shape in the cell mid-plane, and grows in size with the help of microtubules, which dynamically reorganise around its perimeter. The process continues until the plaque entirely forms a new wall, dividing in two the old cell.

References

  1. C. Bustamante, Z. Zev Bryant, B. Steven, S.B. Smith. Ten years of tension: single-molecule DNA mechanics. Nature 421, 423–427 (2003)

    Google Scholar 

  2. A. Ashkin, J.M. Dziedzic, J.E. Bjorkholm, S. Chu, Observation of a single-beam gradient force optical trap for dielectric particles. Opt. Lett. 11, 288 (1986)

    Article  ADS  Google Scholar 

  3. M. Rief, M. Gautel, F. Oesterhelt, J.M. Fernandez, H.E. Gaub, Reversible unfolding of individual titin immunoglobulin domains by AFM. Science 276, 1109 (1997)

    Article  Google Scholar 

  4. K.C. Neuman, A. Nagy, Single-molecule force spectroscopy: optical tweezers, magnetic tweezers and atomic force microscopy. Nat. Meth. 5, 491–505 (2008)

    Article  Google Scholar 

  5. K. Berndl, J. Käs, R. Lipowsky, E. Sackmann, U. Seifert, Shape transformations of giant vesicles: extreme sensitivity to bilayer asymmetry. Europhys. Lett. 13, 659 (1990)

    Article  ADS  Google Scholar 

  6. W. Wintz, H.-G. Döbereiner, U. Seifert, Starfish vesicles. Europhys. Lett. 33, 403 (1996)

    Article  ADS  Google Scholar 

  7. W. Helfrich, Elastic properties of lipid bilayers: theory and possible experiments. Z. Naturfoschung. C: Biosci. 28, 693 (1973)

    Google Scholar 

  8. U. Seifert, K. Berndl, R. Lipowsky, Shape transformations of vesicles: phase diagram for spontaneous-curvature and bilayer-coupling models. Phys. Rev. A 44, 1182–1194 (1991)

    Article  ADS  Google Scholar 

  9. M. Jaríc, U. Seifert, W. Wintz, M. Wortis, Vesicular instabilities: the prolate-to-oblate transition and other shape instabilities of fluid bilayer membranes. Phys. Rev. E 52, 6623 (1995)

    Article  ADS  Google Scholar 

  10. K. Khairy, J. Howard, Minimum-energy vesicle and cell shapes calculated using spherical harmonics parameterization. Soft Matter 7, 2138 (2011)

    Article  ADS  Google Scholar 

  11. N. Gov, S.N. Safran, Pinning of fluid membranes by periodic harmonic potentials. Phys. Rev. E 69, 011101 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  12. M. Reffay et al., Interplay of RhoA and mechanical forces in collective cell migration driven by leader cells. Nature Cell Biol. 16, 217–223 (2013)

    Article  Google Scholar 

  13. H. Jiang, F. Si, W. Margolin, S.X. Sun, Mechanical control of bacterial cell shape. Biophys. J. 101, 327–335 (2001)

    Article  Google Scholar 

  14. L.A. Diaz-Martinez, H. Yu, Chromosome condensation and cohesion, in “Encyclopedia of Life Sciences (ELS)” (John Wiley Ed., Chichester, 2010)

    Google Scholar 

  15. C.B. O’Connell, A.L. Khodjakov, Cooperative mechanisms of mitotic spindle formation. J. Cell Sci. 120, 1717–1722 (2007)

    Article  Google Scholar 

  16. T. Clausen, K. Ribbeck, Self-organization of anastral spindles by synergy of dynamic instability, autocatalytic microtubule production, and a spatial signalling gradient. PLoS ONE 2, e244 (2007)

    Article  ADS  Google Scholar 

  17. C.C. Poirier, W.P. Ng, D.N. Robinson, P.A. Iglesias, Deconvolution of the cellular force-generating subsystems that govern cytokinesis furrow ingression. PLoS Comp. Biol. 8, e1002467 (2012)

    Article  ADS  Google Scholar 

Further Reading

  1. X. Michalet, D. Bensimon, Observation of stable shapes and conformal diffusion in genus-2 vesicles. Science 269, 666 (1995)

    Article  ADS  Google Scholar 

  2. A. Mogilner, E. Craig, Towards a quantitative understanding of mitotic spindle assembly and mechanics. J. Cell Sci. 123, 3435 (2010)

    Article  Google Scholar 

  3. M. Doi, Introduction to Polymer Physics (Clarendon Press, Oxford, 1996)

    Google Scholar 

  4. Alberts, Bray, Lewis, Raff, Roberts, Watson, Molecular Biology of the Cell, 4th edn. (Garland Science, New York, 2002)

    Google Scholar 

  5. A. Mogilner, On the edge: modeling protrusions. Current Opin. Cell Biol. 18, 32–38 (2006)

    Article  Google Scholar 

  6. I. Mendes Pinto, B. Rubinstein, R. Li, Force to divide: structural and mechanical requirements for actomyosin ring contraction. Physical Rev. A 105, 547–550 (2013)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IEMN, CNRS UMR8520, Université de Lille I Sciences et Technologies, Villeneuve d’Ascq, France

    Fabrizio Cleri

Authors
  1. Fabrizio Cleri
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fabrizio Cleri .

Problems

Problems

8.1

Hollow versus filled

Calculate the persistence length and the flexural rigidity, for a hollow microtubule with inner and outer radii 10 nm and 12 nm, and Young’s modulus \(E=150\) MPa. Compare the results with a solid microtubule, with the same amount of mass per unit length.

8.2

Bacterial DNA

The DNA of some bacterium contains 3.45 million base pairs. (a) Calculate the contour length. (b) Given the average DNA Young’s modulus \(E=350\) MPa, calculate its persistence length. How does it compare with the contour length? (c) Given the DNA diameter of 2 nm, what is the smallest volume into which the double strand can be packed? (d) Calculate the end-to-end distance and gyration radius of this DNA. How these compare with the size of a typical bacterium?

8.3

Exocitosis

Find an explicit expression for the surface tension energy, \(E_S = \varSigma A\), during the three steps of the formation of a spherical liposome, illustrated in the figure below. Starting from the membrane at rest (1), a pseudopod is extruded (2), which eventually forms a spherical vesicle of radius R, fully detached from the cell membrane.

figure d

8.4

Membranes with an edge

A cell membrane is composed by two different types of phospholipids, the A with tails of length \(n_A=14\) and the B with \(n_B=18\). By assuming that bilayers form only with lipids of the same kind, and for a lipid-water interfacial tension \(\varSigma =30\) mJ/m\(^2\), (a) what would be the energy per unit length of an A/B interface? (b) what would be the lowest-energy state for 10 domains, each with average surface 100 nm\(^2\), dispersed in a large membrane patch? how distant in energy are the two configurations? (c) could a transition between the two configurations occur because of thermal fluctuations?

8.5

Membranes with a dimple

Consider a thermal fluctuation making a dimple of height h in a patch \(S=\pi L^2\) of membrane. (a) What is the deformation energy of such a dimple? (b) For a bending constant \(K_b=15 k_BT\), at what h / L ratio the dimple could appear by thermal fluctuation of the membrane surface?

8.6

Pulling chromosomes

An average chromosome can be modelled as an elongated ellipsoid, with major diameter \(b\sim 15 \,\upmu \)m, and minor diameter \(a\sim 1\,\upmu \)m. Observe that during mitosis, each chromosome is pulled by the microtubules by some 15 \(\upmu \)m in about 10 min (such values can vary quite a lot according to the cell type). (a) What is an upper bound to the pulling force? (b) How much work is spent, and how many molecules of ATP are used?

8.7

Pushing cells with a laser

Consider a bacterial cell as a sphere of 2 \(\upmu \)m diameter, and an eukaryote cell ten times larger. A typical laser beam of power P used in optical traps can produce a force of the order of a few pN, as approximated by the expression \(f=nPQ/c\), with n the refractive index of the medium, \(Q\sim 1\)% the quality factor of the laser, and c the speed of light. Estimate the power necessary to move the two cells. What does such a calculation suggest?

(*) The terms of the Creative Commons Attribution 3.0 and 4.0 International License (http://creativecommons.org/licenses/by/3.0/, http://creativecommons.org/licenses/by/4.0/) permit use, duplication, adaptation, distribution and reproduction in any medium or format, as long as appropriate credit is given to the original author(s) and the source, providing a link to the Creative Commons license and indicating if changes were made.

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Cleri, F. (2016). Molecular Mechanics of the Cell. In: The Physics of Living Systems. Undergraduate Lecture Notes in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-30647-6_8

Download citation

Publish with us

Policies and ethics

Search

Navigation