# Blood Clot Mechanics at the Molecular Level

## by Andre on 18 January 2007

The function and dysfunction of blood clots are often directly related to their mechanical properties: clots stop blood from flowing through wounds but can also break away (embolize) and block blood vessels causing stroke. Strength and plasticity are both important for ensuring the former is more common than the latter and so people have been studying the mechanics of clots for over 50 years.

Despite this history, or perhaps because of it, new discoveries are being made all the time. Take the recent observations reported by Liu *et al.* last year in *Science* (abstract is free). They used a combined fluorescence and atomic force microscope (not unlike the one I’ve been working with recently!) to stretch single fibrin fibers—the ones that make up the protein mesh of blood clots shown in green in the image above—to see how far they could stretch. They found that some fibers could stretch up to 5 times their relaxed length before breaking! Check out the movies at Martin Guthold’s site.

How can fibrin fibers accommodate so much stretch? For these extreme stretches especially, I think it’s likely that protein unfolding will play a role. But without knowing anything about the mechanics of the proteins that make up fibrin fibers, this is just speculation. And that’s where single molecule stretching comes in. For some background on single molecule mechanics you can see a previous post here. Basically, a protein is adsorbed on a surface and then pulled away using the sharp tip of the microscope’s cantilever. The nonspecific attachment between the tip and protein of interest is sometimes strong enough to pull apart the relatively weak interactions that hold proteins together. To address clot mechanics at the molecular level, I started pulling on fibrinogen, the protein that forms fibrin fibers after activation by an enzyme called thrombin. See last November’s molecule of the month for more details on fibrinogen and the clotting cascade.

Unlike most other proteins studied so far by single molecule AFM, fibrinogen has a more complicated and varied structure and this complicates the interpretation of single molecule experiments. To improve our chances, we took advantage of another enzyme that acts on fibrinogen called factor XIIIa that covalently attaches fibrinogen molecules together. When we mix fibrinogen and factor XIIIa together we get little chains of fibrinogen that are perfect for pulling on:

When these little chains (the scale bar in that image represents 50 nm) are pulled, we see a sawtooth force-extension curve that is consistent with the sequential two-state unfolding of protein domains in series. Before doing this experiment, it wasn’t known *which* domain of fibrinogen would unfold first but, based on the distance between the peaks of the sawtooth and the known structure of fibrinogen, we could rule out the globular end domains and conclude that it is likely the coiled-coil domains that are each unfolding independently. The next figure shows the length increase upon coiled-coil unfolding on the left and the crystal structure of fibrinogen on the right. Notice the rod shape with two end domains separated by a three stranded coiled-coil. It’s quite a looker.

We’ve pulled on fibrinogen and found that its coiled-coils unfold at a force around 100 pN in a two-state like process (the two states being folded and unfolded). This fact, and some elaborations on it, are interesting as pure biophysics, but what does that mean for blood clot mechanics and therefore physiology? Good question, I’m glad you asked. Well, we know the stiffness of single fibrin fibers because of some other work in John Weisel’s lab (~10 pN/nm^2) and we know the approximate cross sectional area of the molecules (~10 nm) so we can calculate that the force per molecule after a two fold stretch of a fiber is around 100 pN, i.e. enough to unfold coiled-coils! This unfolding could serve to absorb just over a two-fold stretch in addition to that available from network rearrangement and fiber bending and that might prevent clot breakage in some cases. But what about at more modest extensions? To be honest, I don’t know, but living with (and reveling in) uncertainty is one of the joys of science. And I have hunches. Hunches are key.

*For some more technical details and references, see our new paper in* Biophysical Journal *available free online here [pdf]*

Congratulations on your paper!

With respect to mechanics of any type, are physicists failing to use a valuable tool of representation theory from applied mathematics?

I have been reading a classic from the Society of Industrial and Applied Mathematics [SIAM]: Tamer Basar and Geert Jan Olsder. ‘Dynamic Noncooperative Game Theory’, revised 1999 from 1982. The authors refer to this as a type of representation theory.

Since this is mathematics, the language is similar, but not identical to representation theory used in physics.

Some differences include using C* for cost-to-come and G* for cost-to-go,

Similarities include index sets, infinite topological structured sets, mappings and functionals in discrete time.

There is substitution for some of these items in continuous time such as time intervals, Borel sets, trajectory, action and informational topological spaces.

Tme appears to be treated as a duality.

There may or not be stochastic influences.

The Isaacs condition for the Hamiltonian is used.

Types of such games include:

for discrete time – OL – open loop

CLPS – closed loop perfect state information

CLIS – CL imperfect state

FB – feedback perfect

FIS – feedback imperfect

1DCLPS – one-step delayed CLPS

1DOS – one-step delayed obsevation sharing

for continuous time – OL

CLPS

eta-DCLPS – eta-delayed DCLPS

MPS – memoryless perfect state

FB

If players are allowed to be entities capable of exchanging enegy quanta or longevity then this might considered enegy economics?

The stochastic game may be consitent with the probablistic nature of QM.

Andre – yes, congrats on the paper!

Doug – ...what?

Hi PhilipJ

My agenda is to encourage those working in physics, especially biophysics, to consider using some techniques from applied mathematics to study mechanics, in various settings, from a game theory perspective.

In reading this paper by GJ Chaitin (IBM Research), ‘Algorithmic information theory: Some recollections’, I noticed that section ‘Challenges for the Future’, states that there is a need for a dynamic rather than static biological mathematic model.

http://arxiv.org/pdf/math.HO/0701164

I think work in this area began shortly after R Isaacs wrote ‘Differential Games I, II, III, IV’ from 1954-1956 or at least in pursuit-evasion games.

This work was continued by many others, including Tamer Basar and Geert Jan Olsder in ‘Dynamic Noncooperative Game Theory’ revised 1999 from 1982 [proofs].

Steven M LaValle, ‘Planning Algorithms’ [PA] 2006, addresses this subject from a game theory, computer programming and robotics perspective, especially in 13.5.2 Differential Game Theory.

http://planning.cs.uiuc.edu/node710.html

There are probably more specific examples for biology than those I provided which are economic and motion oriented. Yet if biology is treated as a type of energy economics then the more general discussion may apply.

John Horton Conway [number theory, game theory, theoretical physics] “in Conway’s provocative wording, if experimenters have free will, then so do elementary particles.”

http://en.wikipedia.org/wiki/John_Horton_Conway

John Forbes Nash Jr, “As a graduate student I studied mathematics fairly broadly and I was fortunate enough, besides developing the idea which led to “Non-Cooperative Games”, also to make a nice discovery relating to manifolds and real algebraic varieties. So I was prepared actually for the possibility that the game theory work would not be regarded as acceptable as a thesis in the mathematics department and then that I could realize the objective of a Ph.D. thesis with the other results.”

http://nobelprize.org/nobel_prizes/economics/laureates/1994/nash-autobio.html

Game theory is applied mathematics!

This editor’s summary from Nature discusses the use of strategies and algorithms in a setting different from fibrinogen. However, this is the type of point that I am trying to make in my inept way of expression. Fibrinogen may use a similar but not identical manner of “infotaxis”.

Nature: Volume 445 Number 7126 pp339-458

Editor’s Summary 25 January 2007 Information trail

Chemotactic bacteria are guided towards the source of a nutrient by local concentration gradients. That works on the microscopic scale, but at larger scales such local cues are unreliable pointers â€” for example, wind or water currents may disperse odours sought by foraging animals. Using statistical techniques, Vergassola et al. have developed a general search algorithm for movement strategies based on the detection of sporadic cues and partial information. The strategy, termed ‘infotaxis’ as it maximizes the expected rate of information gain, could find application in the design of ‘sniffer’ robots

News and Views: Mathematical physics: On the right scent

Searching for the source of a smell is hampered by the absence of pervasive local cues that point the searcher in the right direction. A strategy based on maximal information could show the way. Dominique Martinez

Letter: ‘Infotaxis’ as a strategy for searching without gradients

Massimo Vergassola, Emmanuel Villermaux and Boris I. Shraiman

http://www.nature.com/nature/journal/v445/n7126/edsumm/e070125-10.html