Journal Club Theme of March 1: Measuring Cellular Tractions

Vesna Damljanovic's picture

Cell tractions are the outcome of the complex process of cytoskeletal force generation that cell uses to maintain structural stability, to sense the physical environment and to propel itself.  We are only now beginning to understand the process of cytoskeletal force generation, and we cannot yet say much about the losses in transmission through focal adhesion/integrin complexes (attachment ‘islands’ at the cell-substrate interface), but we can definitely measure the tractions that result from cytoskeletal force generation.  The mechanics behind the measurement method might be of interest to the wider audience of iMechanica, as it involves an interesting inverse problem and different solution methods that have incited lively discussions in past years.

The main idea is to calculate tractions from the measurement of the deformation they caused in a substrate to which the cell is attached. It is assumed that the in-plane forces and out-of-plane displacements are not coupled, and that, effectively, the out-of-plane forces are negligible.

Among the discrete methods, the simplest one uses arrays of microfabricated elastomeric posts whose protein-coated tips act as attachment points for the cells:

Tan JL, Tien J, Pirone DM, Gray DS, Bhadriraju K and Chen CS. Cells lying on a bed of micro-needles: An approach to isolate mechanical force. PNAS 100 (4): 1484-1489 (2003).

Force at the post tip is calculated using Euler-Bernoulli beam theory, but the post's short length (compared to diameter) and considerable deflection do not warrant EB theory’s neglect of transverse shear strain, so using EB overestimates the force.  Also, the inward forces bring post tips in contact with each other and EB theory breaks down—that this happens frequently can be seen in the recent movie [1]. Optimizing the post stiffness to prevent contact while providing measurable deflection, involves repeated expensive microfabrication procedure, which is the main drawback for this technique.

In contrast, the substrate for all the continuum methods is an isotropic linearly elastic solid, usually a hydrogel or a silicon rubber film. Fluorescent markers (beads) are embedded in the substrate surface and their motions are recorded using fluorescence microscopy. Substrate thickness (usually ~100 microns) is at least order of magnitude greater than the maximum surface displacement in order to satisfy the key assumption: that the substrate is a semi-infinite body. This assumption was needed in order to use the Boussinesq Equations, which simplify the solution of this inverse problem. Displacements are given as the convolution of Green’s tensor and traction and the challenge is to solve the inverse problem given the limitation of the noisy displacement field obtained from the measurements.

In the first method that launched the cell traction assays, tractions are explicitly set to 0 outside the cell boundary while the noisy displacement field is specified everywhere, effectively providing for an ill-posed inverse problem that requires somewhat arbitrary smoothing in order to obtain stable solution:

Dembo M and Wang Y-L. Stresses at the cell-to-substrate interface during locomotion of fibroblasts. Biophys J 76: 2307-2316 (1999).

Furthermore, the problem kernel is not diagonal in real space, which leads to bulky matrices and computationally intensive inversion.

In the competing traction calculation method, inversion is performed in Fourier space because the kernel is diagonal in Fourier space and inversion is exact and considerably less computationally intensive:

Butler JP, Tolić-Nørrelykke IM, Fabry B and Fredberg JJ. Traction fields, moments, and strain energy that cells exert on their surroundings. Am J Physiol Cell Physiol 282: C595-C605 (2002).

Lack of periodicity in measured data results in artifact tractions at the boundary of the computational region, but these artifacts are contained only at the boundary and do not affect tractions in the region of interest (cell-covered area) provided that the distance to the boundary is equal to several cell lengths.

When tested on identical data set, the real-space and the Fourier-space method provided reasonably similar results [2].

An interesting detail remains unnoticed by cell biologists: the small-strain assumption is implied in the Boussinesq Equations but apparently unsupported by the experimental data. Substrate stiffness is fine-tuned to allow measurable displacements at given imaging conditions but that results with large strains which can be easily inferred from the time-lapse movies of substrate deformations (strain data is not reported in the references). It seems that the error of small-strain assumption should be considerable but it is not clear if it results in overestimates of tractions, as my intuition suggests, or it is of a more complex nature. iMechanica is perhaps the best forum for discussions of this specific issue and this class of inverse problems in elasticity.


[1] Supporting Movie 1 for: du Roure O, Saez A, Buguin A, Austin RH, Chavrier P, Silberzan P and Ladux B. Force mapping in epithelial cell migration. PNAS 102 (7): 2390-2395 (2005).

[2] Ning Wang, personal communication

Zhigang Suo's picture

What can the biologist do with the measured traction?

When Ning Wang was at Harvard, he explained this problem to people here.  The Fourier analysis sounded excellent. 

It will help many of us biologically intrigued but uninitiated to know the answer to the following question in more detail:  Why does the biologist care about this traction?  That is, with a good technique to measure this traction, what can the biologist do with the data?

Ning Wang's picture

Tractions are fundamental processes of biology

It is nice that Vesna submits Traction as the theme of this month.  This topic is both interesting and funamentally important.

Zhigang asked a question: "what can the biologist do with the data".  It is true that the usefulness of traction data has been underappreciated by biologists, but it is beginning to change.  For years biologists have been keen on measuring those parameters that they think are relevant to biology: cell length, cell projected area, cell shape change, location and movement of a protein, association of a protein with another protein, etc.  Rarely has the question been asked: what drives these processes? With the experimental data accumulated over the last decade from many labs, we may propose that the active contractile forces generated by the cell itself via its myosin motor proteins is the key and fundamental factor in controlling intracellular, intercellular, and cell-matrix movements and

Tractions are the part of those contractile forces that are transmitted to the outside of the cell.  Thus tractions might be crucial in biology for early embryogenesis, development, tissue pattern formation, invasions, etc.  It is anticipated that biologists will gradually realize the importance of tractions in addressing fundamental biological questions such as "how individual cells form a patterned tissue?" and use these data in combination with other biologically and biochemically derived data to obtain a comprehensive understanding of biology.

I would like to raise several other issues for discussion:

  1. The out-of-plane forces are negligible probably only for very spread cells. For less spread cells, these forces may be appreciable.
  2. Traction maps for a cell in a 3D matrix have not been developed.  YL Wang computed tractions for a cell in a "sandwiched" matrix, Ben Fabry started to attack this problem.  But it is rather challenging.
  3. Conventional wisdom is that tractions are generated at focal adhesions.  However, our new experimental data suggest that this may not be true.
  4. YL Wang’s lab recently found that cell migration speed is not altered when tractions are dramatically inhibited, raising the question:  what is the purpose of having such a large traction between a cell and a matrix? (It takes only a very small magnitude of traction to overcome the viscous drag of the fluid surrounding the cell). We speculate that 1) large traction is necessary for cell shape stability, i.e.,cell shear stiffness. We have evidence that cell stiffness is directly dependent on the magnitude of cell tractions; 2) large traction is necessary for deforming the extracellular matrix proteins: M Sheetz lab show fibroblasts generate tractions and deform a collagen fiber; 3) relatively large tractions are generated by the cell to probe the stiffness of its substrate and to adapt its own biochemical activitivies and mechanical parameters such as cell stiffness (mechanosensitivy and feedback loop).

Zhigang Suo's picture

Re: Tractions are fundamental processes of biology

Dear Ning:  Many thanks.  It has always been good to learn from you.  A few followup questions:

  • Issue 2, can you point to a paper or two that describe experimental methods to measure "traction" in a 3D matrix?
  • Issue 3, has your recent work been published?  Can you point to it?
  • Issue 4 goes to the bottom of what I wanted to know:  what can the bologist do with the measured traction?  Hope that you and others can expand this part of the discussion.  Hard facts are good.  Speculations are even better.  A description of possible approaches to study these speculations (or hypotheses?) would teach us uninitiated how you bioloists work.

Ning Wang's picture

Dear Zhigang: Thank you

Dear Zhigang:  Thank you for your probing questions.  I will try my best to address them.  I am sure that you will not be completely satisfied,
but that may be the fun of all these.  Ning

Issue 2: Beningo KA, Dembo M, Wang
of fibroblasts to anchorage of dorsal extracellular matrix receptors.
Proc Natl Acad Sci USA
101:18024-18029, 2004.

Mierke CT, Rosel D, Fabry B, Brabek J.
Contractile forces in tumor cell migration.
Eur J Cell Biol. 2008 Feb 22; [Epub ahead of print]

Issue 3: No.  It is true that focal adhesions do
transmit large tractions.  However, it does not mean other structures do
not.  A 2001 paper in Nat Cell Biol by Balaban et al quantified tractions
only at focal adhesions.  Our work suggests that other dynamic adhesive
structures may generate large tractions too.

Issue 4: There is some nice discussion on the role of
tractions in cell migration in the paper by Beningo KA, Hamao K, Dembo M, Wang
YL, Hosoya H (2006) Traction forces of fibroblasts are
regulated by the Rho-dependent kinase but not by the myosin light chain
kinase.  Arch Biochem Biophys 456:224-31.  Maybe the
primary function of tractions is not for overcoming viscous drag for cell
migration after all.  It is likely that
tractions are used by cells for deforming structures, sensing the physical
environment, and orienting internal and external structures.  It is likely that other important roles of
tractions are yet to be discovered.   


Vesna Damljanovic's picture

Mechanics Aspects

Zhigang , Ning has answered some of your questions regarding biology, so I will not elaborate, otherwise we’ll start a whole new journal club topic.  I was quite keen on a different, mechanics-style discussion—it is up to us, who come to these problems from mechanics background, to provide the accurate tools to biologists, even if it takes them a long time to fully appreciate them. 

 3D case is hard but not impossible experimentally and mathematically, and biologists are quite excited about it hoping to use it to match the ‘physiological conditions’, so yes, it is on my ‘to-do’ list.  But in the situation where we are barely beginning to understand the bits and pieces from the 2D case, 3D case will be exciting but it will take a long time before it can be really helpful because of the added mathematical and experimental complexity. 

So, I believe the greatest challenge this technique is facing right now is to establish how large an error we are making with a small strain assumption in 2D.  Also, how does the noise in the displacement affect this error?  Biologists cannot provide the answers but we can, so I hope I can induce  the people on this forum to discuss it. 

Vesna Damljanovic's picture

Biological Aspects

Regarding biological aspects, it is always hard to make definitive judgment from experimental data in a complex system such as a cell.  There is no guarantee that all the parameters we think we are keeping fixed are really fixed while we perturb the few parameters of interest.  Often the smallest departure from the model renders the original assumption invalid (e.g., Ning’s example of non-zero normal stress for moderately spread cells).  Different experimental scenario often results in opposing results (e.g., correlation between tractions and adhesion size that I mentioned earlier).  What makes it difficult for us mechanicians to quantify is the way biologists utilize perturbations in experiments: inhibit a target protein or complex to understand the mechanism in which this protein is thought to be implicated.  Difficulty is in that this always alters other mechanisms which often affect the mechanism you are trying to characterize.

Also, beeing unfamiliar with mechanics, biologists are hard pressed to simplify it, and often make conclusions that are based on real life experience rather than in-depth knowledge of mechanics laws.  That is how, for example, in the traction assays beginings, migration has been explained as cell front towing the passive cell body and the rear, but in fact, the trailing edge of any migrating cell is generating substantial tractions, is affecting the directionality and magnitude of leading edge tractions and is everything but passive.

So, as I indicated earlier, making definitive conclusions is really hard in this kind of complex system.

I have begun to understand this complexity in my earlier experiments (unpublished) with paxillin (one of the key focal adhesion proteins) and tractions:   it turns out that paxillin-null cells generate equal  or higher tractions than normal cells, but are completely unable to move (these cells are from mouse embryo—due to lack of cell migration ability, these mice die before they complete embryogenesis).  My conclusion was that other focal adhesion molecules take over the transmission of tractions but the adhesion turnover (which is necessary for migration) does not happen because of the lack of paxillin (whose biochemical role in adhesion turnover has been known for some time).

In that vein, I haven’t yet seen Yu-Li’s new work you mention, Ning.  What do you mean by ‘inhibiting tractions’?  There are quite a few molecules that are implicated in traction generation and, as far as I am aware, no single one is the ultimate switch with full on-off ability.   Is it the published paper or work in progress? I would be very interested to hear about it.

Ning Wang's picture

Tractions not important for cell migration?

Dear Vesna, Thank you for
your comments and sharing some of your unpublished results.  Many
people originally believed that the primary purpose of tractions is for
cell migration; that's why people measure tractions during cell
migration.  However, in the paper by Benigno KA and YL Wang et al in
Traction forces of fibroblasts are regulated by the Rho-dependent
kinase but not by the myosin light chain kinase.  Arch Biochem Biophys. 2006 Dec 15;456(2):224-3. Epub 2006 Oct 11, they stated that "The lack of inhibition of cell migration by blebbistatin and Y-27632 [[18], [32] and [33];
unpublished observations], despite the nearly total inhibition of
traction forces, raises serious questions about the biological function
of traction forces, which were previously believed to be involved in
overcoming adhesive resistance and propelling forward migration."

[18] G. Totsukawa, Y. Wu, Y. Sasaki, D.J. Hartshorne, Y. Yamakita, S. Yamashiro and F. Matsumura, J. Cell Biol. 164 (2004), pp. 427-439.

[32] J. de Rooij, A. Kerstens, G. Danuser, M.A. Shwartz and C.M. Waterman-Storer, J. Cell Biol. 171 (2005), pp. 153-164.

[33] W.-H. Guo, M.T. Frey, N.A. Burnham and Y.-L. Wang, Biophys. J. 90 (2005), pp. 2213-2220.

It is now open for discussion: what is the primary role of tractions of a living cell?  I would like to see comments on this.



Vesna Damljanovic's picture

Dear Ning, Thanks for the

Dear Ning,

Thanks for the rerferences.  I was actually traveling on a train while composing these answers off line and didn't see your last post where you answer Zhigang's last question and offer Yu-li's paper as a reference.  You make interesting points that are related to cell signalling and, as a mechanician, I can  take up your challenge only to a certain point.

I have always wondered why "overcoming of viscous drag" has been named as the main (or any) purpose of tractions.  Cells deform collagen matrices in which they are embedded (3D) and there is no viscous drag involved.  The viscous drag, if present physiologically (blood vessels) or in the petri dish,  is primarily matched by the ultimate strenght of adhesions (cohesive zone  @ crack tip in fracture mechanics comes to mind).

Biologists use "traction forces" and "contractile forces" interchangeable (as Yu-Li did in the 2006 paper), but they are not the same thing.  Tractions, as we measure them indirectly, are the final outputs of acto-myosin contraction in the cytoskeleton, but before they 'come out' and are measured by our traction assays,  some of it is lost in transmission through the focal adhesions and integrins. 

I realize this might appear as nitpicking, but because of the above I think we should not use terms such as "inhibit tractions", because we can only inhibit various molecules in the cell and the effect of that can also affect tractions.  So, by using blebbistatin, we only inhibit acto-myosin machinery and we (or better, I) don't know what it does to focal adhesions.

Actually, if blebbistatin dissolves the focal adhesions, then it will make it
easy for the cells to continue migrating despite of lack of acto-myosin
contraction.  But I really ought not to go deeper into biochemistry
issues, as I am not familiar with them.

The bottom line is, blebbistatin inhibits myosin-II so it is understandable that the tractions drop.  Migration, however, needs adhesion turnover as explained in many migration papers (see any review by Horwitz, also check Cell Migration Consortium website), and adhesion turnover is regulated by GTP-binding proteins (Rho, Rac and CdC-42) among other things.  I would still not necessarily conclude that tractions are not necessary for migration: remember that Yu-li's experiment is a perturbation of an already existing steady-state migration.  I am currently working on several methods that would help me look into tractions as they are being generated.   

To summarize, tractions have probably the largest role in maintaining the structural stability of the cell in the specific attachment configurations (single-cell spread out, packed monolayers, tissue).  As a corrolary, tractions are therefore important for maintainging the direction of migration.  Finally, there would be no sensing without the traction mechanism--that is easy to check just by plating the cells on 1/2-stiff and 1/2-compliant substrate as Yu-Li did long time ago in the 2000 Biophys J paper (he didn't try blebbistatin there though).


Ravi-Chandar's picture

Measurement of tractions and speculation on their use in biology

I read this discussion with interest; let me add my two cents worth to this, after warning you that I don't know much about the literature in the field.

In some random searches regarding wrinkling some years ago, I came across the following: Burton et al  (Burton Jung and Taylor, 1999, Keratocytes generate traction forces in two phases. Mol Biol Cell. 10:3745-3769) measured wrinkle lengths on silicone substrates and estimated the tractions. Since they calibrated this with direct measurements using microneedles, I believe measurement accuracy is quite good. Their paper indicates that they were able to measure from a few nanonewtons to a few hundered nanonewtons. I think this circumvents the issues related to the Bernoulli-Euler beam or the Boussinesq solution that are used in the two methods discussed above.

I would like to rephrase Zhigang's question: What does biology do with tractions? Since Zhigang called for speculation, here goes: I like the suggestion that the main role of these tractions may not be locomotion. I suggest that these contact points with the external world are the cell's way sensing the environment and regulating forces that trigger biochemical processes of the cell. Since the maximum force that can be generated is dictated by the substrate stiffness and not the internal workings of the cell, in experiencing these tractions, the cell is sensing the substrate stiffness and hence the environment. There may be some simple ways of testing this hypothesis.

Vesna Damljanovic's picture

wrinkled substrates

Thanks for mentioning wrinkled substrate.  That was the first elastic substrate used before Dembo & Jacobson gave them up, mostly because of painstaking trial-and-error in fabricating the substrate of just the right stiffness to give enough visible wrinkles but not get completely unstable.  Switching to thick gels provided more controlled substrate.

 Problem with wrinkles is that they are non-linear.  Try to calculate tractions from that kind of displacement field, even if you can accurately image it (the deformation)...  There is some interesting work on thin film wrinkling by Gioia in 90's (at UIUC) that could be used but no-one tried.  No wonder, because it would have been more difficult and computationally involved than Boussinesq stuff.

 The only measurement I saw so far is that of geometric characteristics of the deformation followed by 'calibration' for some generalized force.   This method cannot capture the local gradients and, most importantly, cell deforms the substrate so much that the wrinkles themselves affect the subsequent force direction (analogously to the micro-needles).  Keratocytes are wonderful model cells in that they have a very symmetric and reliable force field, so they may not be affected so much, but fibroblast or any other unordered cell will be adjusting the forces to match the wrinkles (perpendicular to them). 

Ning Wang's picture

Tractions are for shape stability/substrate rigidity sensing

It appears that some of us would agree that primary known functions of tractions are for cell shape stability (our papers in 01, 02; Discher papers, 04, 06, etc) and substrate rigidity sensing (YL Wang papers, Discher papers, Janmey papers, etc).  As to cell migration, it is just so peculiar that the cell would generate such an unnecessarily high magnitude of traction for cell migration.  In YL Wang's 06 paper, "inhibition of tractions" simply states the fact that tractions are much lower than controls after various interventions.  We will see how this issue raised by YL Wang's lab will be addressed in the future by other labs.

Vesna Damljanovic's picture

Traction distribution not magnitude is important for migration

Ditto, Ning.

 Also, you wrote: "As to cell migration, it is just so peculiar that the cell would generate such an unnecessarily high magnitude of traction for cell migration."

Cells are definitely not generating tractions specifically for migration.  My old experiment I mentioned shows that clearly: when you knock-out paxillin (key migration-related protein), it completely kills migration, but the cell is still applying tractions, often higher than a migrating cell. 

Tractions are always generated by a normal adherent cell. 

It is the distribution of the tractions between the rear and the front--not the absolute magnitude--that is 'used' in migration by setting up the direction of migration.

Incidentally, directionallity is pretty important but apparently quite hard to characterize due to lack of good reference due to large 'deformation' of the cell shape, at least for cells like fibroblasts (myocytes and keratocytes, for example, maintain the self-similar shape).  That is another interesting topic.

The magnitude of in plane traction

The discussion is really nice. I have two questions on the magnitude of traction in cell cytoskeletal, maybe Wesna, Ning, or others can shed light on them.

In Tan et al.'s experiments, two types of traction in focal adhesion is reported, with one (in focal adhesion spots of size less than 1 micron^2) much higher than the other (larger focal adhesion size with a traction about 5.5 kPa). Others only observed a maximum traction in the range of 5.5 kPa (Dembo and Wang, Balaban et al., du Roure et al.). Is there any biologic explanation on the differences in these observations? Also, what is the biological process to maintain a saturated traction on the order of 5.5 kPa? 

Many thanks.


Teng Li's picture

Matrix stiffness directs how human stem cells differentiate

I've been following this interesting and informative thread of discussion. Here are some aspects I'd like to learn more:

Ning mentioned that one of the primary known function of cell traction is to sense matrix rigidity. I came across the following paper showing, by changing the matrix stiffness, human mesenchymal stem cells can be directed along neuronal, muscle, or bon lineages.

Matrix Elasticity Directs Stem Cell Lineage Specification.

Cell,Volume126,Issue 4,Pages 677-689



The paper also showed an approximately linear relation between the cell stress (traction) and the matrix stiffness.  My question is, have similar behaviors reported in this paper been observed in other types of cells? For example, when cells sense the rigidity of the matrix, how do they respond in terms of traction, intracellular strain? Any mechanical quantities of the cells possibly remain unchanged?  

Thanks for your comments. 

Ning Wang's picture

Cell stiffness is a key parameter in substrate stiffness sensing

Teng, It is nice that you have an interest in this fascinating field.

We first reported the linear relationship between cell traction and cell stiffness in human airway smooth muscle cells and endothelial cells (Wang et al, PNAS, 2001, Am J Physiol Cell, 2002). Discher's lab then found similar relationship in skeletal muscle cells (Biophys J, 04).

YL Wang first reported  that tractions increase with substrate stiffness in fibroblasts (HB Wang et al, 2000).  Discher et al found the same behavior in HMS cells (Cell, 06).  Recently PA Jamney lab showed cell stiffness increased with substrate stiffness (Biophys J, 07).

With all these findings, one may conclude that cells match their stiffness with substrate stiffness via the action of tractions, that, in turn, is controlled by the prestress (active contractile myosin-dependent forces).  The molecular and biochemical mechanisms are not clear yet.  But one big picture is beginning to be clear: cell stiffness/cell traction is a key parameter in substrate stiffness sensing.  Maybe to regulate intracellular strains?


To Yujie: those high tractions have been reported by YL Wang's lab in newly-formed focal complexes.  The biological significance is not clear.

Vesna Damljanovic's picture

Thanks for references

Thanks for references Ning.

I'd add to Yujie: as I pointed out earlier, Tan et al used micro-needles which affect where and how cell forms adhesions, so I wouldn't compare that experiment with the others.  And just in general, different types of cells generate different levels and distribution of traction.  Fish scale keratocytes are very fast migrating cells with quite weak tractions distributed side-to-middle (almost perpendicular to the direction of motion), while fibroblast are quite slower but with considerable tractions that are oriented front-to-back (axially). 

Thanks to Vesna and Ning

Thanks to you for the input. It helps a lot.