An exploration of dynamic force modeling in cells applied to unexplained results in the Nagayama cell traction investigation.

1. Introduction

Much valuable research in recent years derives from improved capabilities to analyze dynamic forces in cells. These permit us to broaden our understanding of inter- and intracellular pathways, and to deepen our comprehension of phenomena that arise as emergent properties of cellular dynamic forces. Biopolymer science offers an exemplary model of this in the form of research into the dynamic generation of forces by microtubules (MTs) through polymerization and depolymerization.

2.1 Brownian ratchets

Advances in resolution at the nano-scale have aided in the understanding of how microtubules generate pushing forces by adding tubulin dimers to their ends between themselves and the load they are pushing. This is modeled by considering the load to be particle-like, or wall-like. In either case “pushing” is not modeled as a continuous push over time, but rather a probability distribution of interactions that are driven by a “Brownian ratchet” mechanism—that is, once the load has been advanced a certain distance, the addition of a tubulin dimer to the end of the growing microtubule denies the load from retreating to occupy the space now filled by the new dimer, and so movement is one-way and inexorable—hence the ratchet. In the case of a particle like load, the probability distribution refers to the position of the load, as it explores a small region of space in front of the microtubule, though large enough to accommodate polymerization—hence, Brownian. In the case of a wall-like load, it is the microtubule itself that explores a region of space enabling triphosphorylated tubulin dimers to access its active site (Hedde, 2014); these systems are sometimes referred to as elastic Brownian ratchets.

It is possible to express the velocity by which a load can be pushed: 

where v is the velocity, F the load, μ is the change in microtubule length with respect to the addition of one dimer, kon and koff  are the rates at which dimers add on and off respectively, kB is the Boltzmann constant, and T is the temperature in Kelvin (Gittes, 1996). Less intuitive is the pulling activity of microtubules, in which microtubules pull by depolymerization. According to Biased Diffusion Theory, the load binds to a site on a microtubule that serves two functions: first, it permits it to drift along the microtubule in either direction by successive binding of the same site on adjacent dimers, and second, it doesn’t permit the dimer to dissociate if it’s the last one (Keener, 2014). In this way another Brownian ratchet is established. A useful mathematical expression for the speed at which a microtubule can pull a load, is 

where xi s the position of the load at the beginning of period τ, and xi + 1 is the position at the end of the period, Fi is the load at the beginning of τ as a function of x, γ is a frictional coefficient, D is the diffusion coefficient, and pi is a random number selected from a normal distribution with a mean of zero and standard deviation of one (Grishchuk, 2012).

2.2 Multiple applications

An improved understanding of the dynamic forces of microtubules has aided in the elucidation of certain mechanisms of mitosis and the dynamics of the mitotic spindle. For example reconciling the understanding of the front-loading nature of microtubules with the rear-loading mechanism of F-actin revealed how these two polymers counterbalance one another during with F-actin growing apically and microtubules growing in the plane of the epithelium (Woolner, 2012).

Of course dynamic forces at the cellular scale comprise much more than linear pushing and pulling models. Modeling the relationships between cell adhesion and cell deformation are another important area of research. In particular, modeling of cell capture, rolling and detachment as functions of sheer, deformation, and binding affinity stand to improve the efficiency of selective capture in microfluidic antibody systems, which becomes exceedingly important when trying to capture rare species in whole blood (Luo, 2013).

3.1 The Nagayama investigation

In an experiment carried out by Kazuaki Nagayama at the Magaya Instityte of Technology, in which cells were adhered to an elastic micropillar substrate that was mechanically stretched, which stretched the cells in turn by the focal adhesion points where their plasma membranes interacted with the micropillar tips, it was observed that during the first cycle of stretching, 55% of the cells actively recovered, while the remaining 45% recovered passively. Furthermore, when they stretched detached cells with micropipettes, none of the cells exhibited active recovery. The traction forces exerted on the cells were evaluated by plotting the change in position of each micropillar tip with respect to its base (which could not be observed, but which could be calculated by the positions of unloaded pillars) as the substrate was stretched through 50 μm, at an incremental rate of 1.5 μm every 15 seconds at 37 °C. The magnitude of the traction force at each micropillar was calculated using the geometry of the pillars and the Young's modulus of the substrate material to derive its spring constant, and taking the product of this with the magnitude of micropillar tip-deflection. Cells were selected such that their major axes did not exceed 30° from the applied force vector. For each cell the corresponding micropillar deflections were mapped to its major axis, and the mean magnitude of forces calculated along this vector was taken as the cell's total traction force (Figure 1). It was in plotting these values Nagayama observed the active and passive recovery phenomena. These manifested as clockwise plots in the case of passive recovery, since the traction force after the stretch cycle was less than the pretension force, and counterclockwise plots in the case of active resistance, as the final traction force was higher than it was it its pretension state (Nagayama, 2011).

3.2 Constructing a simple model

Let us now apply some of the previously discussed theory to attempt, by means of a simple model, to explore a possible explanation for the unexplained Nagayama results.

We can broadly identify three possibilities:

     I.    Stretching the cells can potentially hinder their ability to recover actively.
     II.   Stretching the cell can potentially trigger active recovery.
     III.  The phenomena are unrelated.

Forced deformations have the potential to damage cells and so active recovery from a forced expansion or forced compression is a potentially selectable trait, so let us explore the idea that proposal 3 is not the case. Proposal 2, requires that the process of active recovery has a resting state of "off," and that it was switched on in slightly more than half of the stretched cells. If this were the case, then we would expect to see one of two responses to the second round of stretching, depending on whether the recovery mechanism switched off before the second stretching cycle can commenced. If it did, then we would expect the mechanism to be switched back on in approximately half the cells since the conditions between the two cycles were essentially identical and the mechanism turned on with 55% probability in the first cycle. On the other hand, if the mechanism stayed on, then we could expect more cells to undergo active recovery in the second cycle if the mechanism continues to turn on with 55% probability. This line of reasoning reduces what is ultimately a complicated biological mechanism to a reliable binary actor, however we can see that for the purposes of a constructing a simple model, proposal 2 is untenable since a simple interpretation of it predicts the same or more active recovery in the second stretching cycle, while Nagayama observed no active recovery. 

Let us now turn our attention to proposal 1, which requires that the resting state for active recovery be "on," and that the mechanism by which active recovery is achieved can be impaired by stretching.

From the material discussed in 2.1, we are aware that microtubules have the potential to exert pulling forces.  Let us simplify our view of the cytoskeleton for the purposes of our simple model and take all it's components to be microtubules bound to microtubule associated proteins (MTPs) embedded in the interior aspect of plasma membrane rafts. We can use equation (2) to derive a stall load, Fs

And so any force greater than Fs will be sufficient to inhibit active recovery. This seems promising, since it is easy to imagine the applied stretching force exceeding the stall load, however, at the end of the cycle the applied force goes to zero, at which point all cells should exhibit active recovery.  Unless, of course, the cell sustains damage during the stretching cycle. Let us develop this thought.

Consider the orientation of a cell that casts an elliptic shadow on the plane of the substrate, and the two extreme cases:

     1)    The cell's major axis is orthogonal to the direction of stretching.
     2)   The cell's major axis is parallel to the direction of stretching.

Now let us consider an elastic micropillar substrate plane upon which a population of cells have been permitted to adhere. For each cell, let N microtubules be interacting with the plasma membrane at points where MTPs are anchored to the inner membrane surface, and let us assume the 3D orientations and lengths of these microtubules is randomly distributed. Observe that stretching the substrate will transmit stress to the plasma membrane of each cell at adherence points (the points where its membrane is adhered to the substrate), and some portion of this stress will be communicated intracellularly to the membrane-anchored microtubules in the form of tensile forces. Note that as the substrate is stretched along some axis, the greater the distance separating any two adherence points on either side of an arbitrary line normal to the stretching axis, the farther they will become separated for any unit quantity of stretching, and so the greater the tensile stresses that mcrotubules spanning these distances will be exposed to during the stretching cycle. Discretizing the adherance points by equating them to their corresponding pillar loci, this can be expressed

where Δx is the change in distance between the pillars, S(F) is the amount each molecular unit of the substrate spreads as a function of tensile force, N is the number of molecular units separating the pillars, and d is the distance from the closer pillar to the transverse axis, and mean l is the mean molecular unit length of the substrate.  For the sake of simplicity, let us impose the condition that S(F) and F scale linearly with slope s, such that

Note that the case 2 cell crosses a wider range of pillar distances d, than does the case 1 cell. It follows then that the sum of the Δx values for the case 2 cell must be higher than those of the case 1 cell, provided the initial positions on the substrate for both cells is the same (or to generalize, normally distributed).

Now let us consider the affect of the substrate stretching on our model of the cell's cytoskeleton. Let us use Hook’s law to express the tensile force required to disassociate an MT of length δ from its anchoring MTPs in terms of a force-loading rate:

where B and E are the bending and elasticity moduli of the MT, respectively. Taking the derivative of (5) with respect to time, and substituting into (6), we arrive at an expression for the change in distance (parallel to the axis of stretch) over the interval τ, which, if exceeded, will result in the dissociation of the MT from one of its MTPs

where the change in δ is

Given that

we observe that the probability of MT/MTP dissociation in the case 2 cell is greater than that in the case 1 cell, and so the probability of an active traction force response to stretching would be greater in case 1 than in case 2. Furthermore, it can be reasonably speculated that for a constant major axis length there exists a gradient of dissociation rates between these extremes, and that there is some critical angle of orientation that establishes the minimum divergence from the stretch axis that permits active response after stretching, and that this angle must be less than 30°, given that this was the maximum angle of orientation permitted in the Nagayama investigation.

However, if the force triggering the MT/MTP dissociation is a direct result of the stretching force, and two identical stretching cycles were performed, we would expect all (or very nearly all) MT/MTP dissociation events that could occur in the first cycle would occur, and approximately zero dissociation would occur in the second stretch cycle.  But at the molecular scale, the actual force triggering the MT/MTP dissociation derives from random bombardments by molecules in the cytoplasm (which are a function of temperature and the duration of the stretch cycle) and not the stretching force. We would therefore expect a probability distribution of events to occur, and so some couplings might survive the first cycle, but not the second—i.e., we would expect dissociation events at both stretch cycles. But for each subsequent stretch cycle the coupling survival probability would left-shift as the MT/MTP coupling population is reduced, so the number of dissociation events at iteration n, is expected to be greater than that of iteration n + l.

3.3 Comparison with experimental data.

This may go some way toward explaining the phenomenon in the Nagayama investigation, however the major axis length and orientation were not presented with their findings. That said, this model is consistent with their finding that nearly half of the cells exhibited a clockwise rotation in the plot of total force as a function of cell strain on stretch cycle 1 (indicating a passive response to stretching), while the remainder exhibited plots with a counter clockwise rotation on stretch cycle 1 (indicating an active response to stretching), and for all cells the difference between the initial and final traction force was significantly larger for cycle 1 than for cycle 2 (Figure 2).

Let us now consider differences that arise between substrate stretching and micropipette stretching. At the moment substrate stretching commences it is only "felt" on the ventral aspect of the plasma membrane, as lipid bilayers have high elastic moduli. In fact, lipid membranes exhibit viscoelasticity and so the elastic modulus only dominates for a very short interval: as long as it takes for the stress to be communicated to adjacent membrane regions of lower tensile stress from which lipids will flow to accommodate the imposed deformation (on the order of tenths of nanoseconds). This rearrangement of membrane lipids is controlled by the membrane's viscosity modulus, and will be largely confined to those existing in non-raft domains. Therefore, if the positions of dorsal raft domains were labeled and tracked, the dorsal aspect of the cell would initially appear to undergo a paradoxical contraction as the number of lipids between dorsal rafts diminishes.  Nevertheless, the cell is increasing in area in the plane of the substrate and so strain is soon communicated to the dorsal membrane region.  However, this strain can be dissipated by decreasing the cell's height normal to the plane of the substrate, and because the viscosity modulus of the membrane is greater than that of the cytoplasm, the modulus of the membrane is rate limiting and so cytoplasmic resistance to this dissipation is trivialized.

Of course, there must come a point at which the strain of the dorsal membrane region approximates that of the ventral region, however the time lag involved in the mediation of the transmission of strain by the membrane's viscoelastic properties is short but non-trivial.  Recall that one of the parameters controlling the probability of MT/MTP dissociation is time (the more time the coupling is subjected to tensile forces, the greater the probability of a cytoplasmic impact with sufficient energy to disassociate the two species).  Therefore, however many MTs remain bound to two MTPs anchored in the ventral region membrane throughout a stretch cycle, it is probable that the number of those surviving in the dorsal region of the plasma membrane will be greater.

Let us now consider the situation with respect to a detached cell exposed to tensile stretching forces by means of a pair of micropipettes adhered at arbitrary poles.  No quantitative data was reported on this phase of the Nagayama investigation, and so it is unknown how the applied force, stretch distance, or loading rates compare to those of the substrate bound cells.  As a thought experiment, however, we can explore what might comparatively happen in the case where the pipettes adhered to the cell are separated at the same rate and magnitude as in the substrate case. 

Note that in equation (1), the largest possible value Δx can have is the change in distance between the clamps affixed to the ends of the substrate, that is no two points on the substrate between the clamps on its ends can separate more than the clamps themselves, which is intuitive.  Therefore, Δx clamp ⋙ Δx cell, given that the micropipettes axis of stretching defines the major axis of the cell, it follows from previous reasoning that under these conditions, we can expect a significantly higher probability of MT/MTP dissociation in the micropipette cell, then in the case 2 substrate cell, and so if the experimental conditions allowed for any substrate adhered cell to demonstrate a passive response to the first stretch cycle, we would expect the micropipette cell to exhibit this attribute as well.  Let us now consider the case where

Δx clamp = Δx cell.  Now we have a situation in which interactions between the cell's focal adhesion points and the micro-pillars has caused the substrate cell to flatten out and so can be expected to have a greater surface area at its pretension resting state than the micropipette cell, which will be more spherical at its pretension state.  And while the substrate cell dissipates strain from an increase in lengths by a decrease in height (1 axis), the micropipette cell can readjust its volume along the transverse plane (2 axes) and so the inner surface area will increase at a slower rate than that of the substrate cell. Therefore we expect the probability distribution of MT/MTP dissociation to be greater in the substrate bound cell. This situation would give rise to an increased probability of active response of the micropipette cell after the first stretch cycle—except in the case that the pretension approximates its resting (spherical) state.

4.1 Final remarks

Without additional experimental data, this is as far as it is reasonable to take this exercise.  The object was to illustrate the potential utility of applying a small amount of the insight derived from the study of dynamic cell forces to a seemingly mysterious phenomenon.  As it happens the phenomenon arose in a study of dynamic cell forces, but this just underlines the value in a holistic approach, as the model we developed considered different dynamic forces than those under investigation in the study.