Wednesday, April 22, 2009

Adaptive fiber networks

Consider a model cell in the "spider" geometry: A central cell body with N radial stress fibers, each fiber starting at the cell center and terminating at the remote end in a focal adhesion contact to the substrate. (Compare figure).

Let us assume that the fibers can be modelled as linear springs of stiffness k, however with a time-dependent rest length L0(t). The force is at any time given by F=k*(L-L0), if the actual fiber length L exceeds the rest length L0. The force is zero otherwise (stress fibers don't withstand much compression).

Now assume the fibers are adaptive. They have a preferred (or "goal") prestress F_goal, and they change their rest length in order to come closer to this goal. For simplicity, assume a constant adaption velocity in both directions:

dL0/dt = +v if F>F_goal (lengthening)
dL0/dt = -v if F<F_goal (shrinking)

Birth of fibers: The fibers are born with fixed initial length L(t=0) and with random directions. There are available "slots" for only NMX fibers in total. The birth rate of the fibers is proportional to (N-NMX) and therefore self-limiting. At time of birth, a fiber is relaxed, i.e. L0=L(t=0).

Death of fibers: Assume that fibers die if their length falls below L_min (under-length) or exceeds L_max (over-stretch). Dead fibers immediately dissappear.

What would be the expected behaviour of that model ?

Remove all fibers from the model cell and place it onto a flat substrate with friction. After some time, a first new-born fiber will appear (say, at the right side of the cell) and adhere to the substrate with its remote end. Being initially relaxed, it will start to shrink. This generates a traction force on the cell, which is thus pulled to the right side against the friction force.

Let's say that before the focal end point is reached, another fiber is born at the left side of the cell. It also starts to shrink, making it harder for the right fiber to increase tension and to reach the goal prestress. So, both fibers work against each other. The cell will now be the node of a spring network with time-dependent prestress and perform some "complicated" motion.

If not disturbed by further birth processes, the cell might reach an euqilibrium position, with both fibers in their goal prestress state.

What if the second fiber also appears at the right cell side ? Then both fibers shrink until they die from under-length. Fibers need some resistence to develop their goal prestress. This resistance can be provided by other fibers or by some external forces (if the cell is sitting in a mechanical potential well).

Consider next an initial situation with N fibers, all in their ideal state (F=F_goal).

If we quickly displace the cell center from its equilibrium position, we shall experience spring-like restoring forces from those fibers that are stretched, zero forces from those that are compressed. All in all an elastic response (With some viscous contributions).

If the displaced cell is quickly released, and if the displacement was not so large to cause the death on any fiber, the cell will return to its equilibrium postition because there was not enough time for adaption.

But if we hold the cell for a long time in the displaced position, the fibers will adapt their rest lengths: The stretched ones will lengthen, the compressed ones will shorten. As a result, after releasing the cell it will return to a new place in between the former equilibrium position and the externally imposed position (plastic deformation).

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