^{*}

Conceived and designed the experiments: MK IVM. Performed the experiments: MK. Analyzed the data: MK IVM. Wrote the paper: MK IVM.

The authors have declared that no competing interests exist.

T-killer cells of the immune system eliminate virus-infected and tumorous cells through direct cell–cell interactions. Reorientation of the killing apparatus inside the T cell to the T-cell interface with the target cell ensures specificity of the immune response. The killing apparatus can also oscillate next to the cell–cell interface. When two target cells are engaged by the T cell simultaneously, the killing apparatus can oscillate between the two interface areas. This oscillation is one of the most striking examples of cell movements that give the microscopist an unmechanistic impression of the cell's fidgety indecision. We have constructed a three-dimensional, numerical biomechanical model of the molecular-motor-driven microtubule cytoskeleton that positions the killing apparatus. The model demonstrates that the cortical pulling mechanism is indeed capable of orienting the killing apparatus into the functional position under a range of conditions. The model also predicts experimentally testable limitations of this commonly hypothesized mechanism of T-cell polarization. After the reorientation, the numerical solution exhibits complex, multidirectional, multiperiodic, and sustained oscillations in the absence of any external guidance or stochasticity. These computational results demonstrate that the strikingly animate wandering of aim in T-killer cells has a purely mechanical and deterministic explanation.

Beyond the more widely known molecular recognition of antigen, specificity of the cellular immune response relies on the precise orientation of immune cells toward infected and tumorous cells. We studied the mechanics of the structural orientation of T-killer cells (a type of immune cells) to their immunological targets. One of the most remarkable features of this process as seen under the microscope is the apparent “wandering of aim”: instead of pointing steadily at the intended target, the killing apparatus inside the T-killer cell can wave around. When two targets are engaged simultaneously, the killing apparatus in the T cell can repeatedly oscillate between the two. It might appear that the origin of this strikingly animate behavior should lie in stochasticity of the underlying mechanism. Our numerical model, however, was able to reproduce the complex, continuing motion in spite of the fact that the model was purely deterministic. This result suggests that deterministic quantitative explanations and supporting experimental evidence can be sought in the other cases of extremely complex cell motility that give the microscopist an acute sense that the object is alive.

The high specificity of the immune response depends in large measure on direct
cell-cell interactions. An example is the interaction of a T-killer lymphocyte with
a tumor cell, or with a cell that has been infected and is producing new viral
particles. It is generally accepted (e.g., ref.

The killing apparatus in T cells is structurally assembled around the centrosome, the
organelle in which the microtubule fibers of the cytoskeleton are anchored.
Experiments suggest that the killing apparatus may be positioned next to the target
cell by molecular motors. According to this hypothesis, dynein motors anchored at
the T cell interface with the target “reel in” the centrosome by
pulling on microtubules that pass over the interface

Is the pulling mechanism biophysically plausible? And what is the nature of the apparent wandering of aim in T-killer cells? Here we show by means of biomechanical modeling that the pulling mechanism is indeed capable of bringing about the functional orientation of the centrosome in a range of conditions. Our analysis also predicts substantial and verifiable limitations of this mechanism. Our calculations show that the complex fluctuations are an intrinsic property of this mechanism and of the T-cell structure, in the absence of any stochasticity or external guidance, suggesting a deterministic mechanical explanation for one of the most “animate” cell behaviors.

From the experimental videos

The cartoon depiction of the dynein motor molecules (red) is for visualization purposes only. Individual dynein molecules are not modeled computationally, only the pulling force they produce. Microtubule thickness is greatly exaggerated in the diagram. The centrosome (green) is merely a marker in the diagram; the centrosome in the model is identified with the common anchoring point of the microtubules.

The active microtubule sliding in the model is meant to represent the action of
cortically anchored molecular motors. Idealizing what should happen when
microtubules come in contact with the cell cortex on which motor molecules are
anchored

Dynamics of the centrosome orientation in a T cell developing
sequentially two synapses is shown. The insets are computer-generated
snapshots of the actual numerical model cell. The graphic conventions
are the same as in ^{2}.

The orientation of the centrosome is described here using an angular measure. The
rounded outline of the T cell makes the angular measures and the terms
“orientation” and “reorientation”
convenient. It also makes the centrosome trajectory during the long-range
reorientation look at least partly like an arc. To show as much of this movement
as two-dimensional representation can convey, we chose throughout our paper to
show reorientation in figures and videos from such an angle that the line of
sight is directed along the axis of the arc. From any other angle, the same
movement would appear only less arc-like, and more
“vectorial”. In this sense, we feel that our model is
compatible with the vectorial description of translocation in experiments

The movements in our model are, strictly speaking, a superposition of the
movements caused by pulling and of movements caused by the deformation of the
cell outline in the beginning of each simulation. Simulations in which pulling
force density was set to zero (

Given the quasi-exponential kinetics of the reorientation to the target, i.e.,
one characterized by a rapid beginning followed by a slow stabilization at the
final position (

(A) The time it takes the model centrosome to reorient by one-half of the
initial angular separation, as a function of this initial separation,
plotted for the indicated values of the microtubule length. The segments
of the broken lines connect the points corresponding to the actual
simulation results; where the segments are dashed, it indicates that
they connect two data points between which a data point is missing
because the half-reorientation could not be achieved. Pulling force
density, 40 pN/µm; effective cytoplasm viscosity, 2 pN
s/µm^{2}. (B) Qualitatively different predictions
obtained with the different microtubule length and initial angular
separation between the centrosome and the middle of the synapse. Regions
in the two-dimensional parameter space are color-coded and numbered. In
region 1, the complete reorientation is achieved. In region 2, the
reorientation is “jammed” at around 30° of
remaining angular separation. In region 3, the reorientation is
“jammed” at the characteristic angular separation of
100°. In region 4, reorientation does not commence because the
microtubules are too short to contact the synapse. In region 5, complete
reorientation is achieved after a catastrophic stability loss of the
“locked” configuration of antiparallel microtubules
overlapping at the synapse. In region 6, the same happens but the final
reorientation is as incomplete as in region 2. In region 7, the
“locked” overlapping configuration is stable and no
reorientation occurs. Pulling force density, 40 pN/µm;
effective cytoplasm viscosity, 2 pN s/µm^{2}. (C)
Effect of microtubule dynamic instability on the stability of the
“locked” configuration such as predicted in region 7
of (B). Angular position of the centrosome is plotted vs. time as
predicted by the purely deterministic model analyzed throughout the
paper (black curve) and with an additional assumption of stochastic
microtubule dynamic instability (colored curves). The three stochastic
simulations are independent (in the sense of pseudo-random number
generation on a computer) repetitions of a simulation which was
otherwise set up the same way as the deterministic one. The angle
plotted is defined as the angle formed by the lines drawn from the
nucleus center to the centrosome and to the middle of the synapse. The
deterministic prediction is that the centrosome, having started facing
the opposite side of the cell from the synapse, will not be able to
reorient to the synapse. The stochastic predictions differ between runs:
one is similar to the deterministic prediction, in the other two the
centrosome was able to reorient. Pulling force density, 40
pN/µm; microtubule length (starting microtubule length in
stochastic simulations), 21.5 µm; effective cytoplasm
viscosity, 2 pN s/µm^{2}.

The complexity of outcomes reveals the limitations imposed by the basic cell
structure on the functional capacity of the pulling mechanism. The chart of the
simulation outcomes (

Additional simulations where dynamic instability

The mechanically dead-locked state with the non-functional orientation of the
centrosome has not been experimentally documented. This suggests three
possibilities: (1) the specific initial conditions that lead to it in the model
(region 7 in

Returning to the analysis of the purely deterministic effects of pulling (without
incorporating the dynamic instability of microtubule length in the model), we
analyzed further the mechanism of the deterministic mechanical instability of
the centrosome position that followed the long-range reorientation.

(A) Graphs of the model cell structure at the indicated time points. (B)
The oscillating microtubule system shown in projection onto the synaptic
plane. The parts that are in contact with the synaptic surface and are
experiencing the pulling are highlighted in red. Pulling force density,
20 pN/µm; microtubule length, 16 µm; effective
cytoplasm viscosity, 2 pN s/µm^{2}.

(A) A centrosome trajectory in projection onto the synaptic area, with
color denoting the height above it and arrows, the direction. The
directions of axes are as indicated in ^{2}. (B) Positions of
the centrosome along the two horizontal axes and its vertical position
plotted vs. time. Note the phase shift between the oscillations along
the ^{2}. (C) Effect of
microtubule dynamic instability and of an annular shape of the pulling
surface on the pattern of oscillations. Position of the centrosome is
plotted vs. time as predicted by the purely deterministic model with the
disk-shaped pulling surface, as analyzed throughout the paper (black
curve), and with stochastic microtubule dynamic instability and annular
pulling surface (colored curves). The two stochastic simulations are
independent in the sense of pseudo-random number generation on a
computer. The stochastic predictions differ between runs but preserve
the characteristic features of the deterministic one. Pulling force
density, 20 pN/µm in the deterministic simulation and 36
pN/µm in the stochastic simulations. Microtubule length
(starting microtubule length in stochastic simulations) was 16
µm, effective cytoplasm viscosity, 2 pN
s/µm^{2}.

To determine the impact dynamic instability of microtubules

As regards the origin of the deterministic oscillations and of the repeated
overshooting which are exhibited by the centrosome, it is important to point out
that inertia plays no role in intracellular movements due to the prevailing
near-zero Reynolds number conditions. In fact, like in models for comparable
types of intracellular movements (e.g., refs.

Simulations with different pulling force densities show that the basic frequency
of the oscillations is fairly insensitive to this parameter, although the
overall pattern of oscillations changes abruptly when a certain value of it is
crossed (

(A–C) The three types of oscillations that are predicted
correspondingly with low, intermediate, and high values of the pulling
force density. The centrosome trajectory is plotted in the
^{2}. (D) The mean
period of oscillations parallel and perpendicular to the synapse, as a
function of the pulling force density. The error bars are S.E.
(insignificant in size for most data points). Microtubule length, 16
µm; effective cytoplasm viscosity, 2 pN
s/µm^{2}. (E) The mean (solid line) and the
characteristic minimum and maximum (dashed lines) of the centrosome
distance from the synapse, as a function of the pulling force density.
The minimum and maximum attained during each period were averaged over
many periods to obtain the values of the minimum and maximum that are
characteristic of the given force density. The error bars in this plot
show the standard error associated with the statistical estimation of
the characteristic minimum and maximum values. Microtubule length, 16
µm; effective cytoplasm viscosity, 2 pN
s/µm^{2}. (F) The peak deviation of the centrosome
from the midpoint (amplitude) in oscillations parallel to the synapse
(^{2}.

Capacity to explain oscillations of the centrosome within a synapse is a
stringent test of a mechanism proposed for centrosome polarization, and our
computer simulation results indicate that the empirical hypothesis of cortical
dynein pulling

Numerical solution shows that after simultaneous development of two synaptic
areas on two sides of the initial centrosome position, the model centrosome goes
to one of them. Which one it goes to first in our deterministic model can be
decided by an otherwise insignificant deviation of the initial centrosome
orientation from the middle, such as by 2°. What is important is that
after pausing at the first synapse, which pause can last for a significant
period of time, the model centrosome spontaneously moves to the other synapse
(^{2}, respectively), it is easy to reproduce with
remarkable precision both the duration of the pause and the duration of the
movement phase (

(A) Graphs of the model cell structure. The angle between the two
synaptic planes is indicated by the red arc and equals 144° in
this simulation. Pulling force density, 40 pN/µm; microtubule
length, 16 µm; effective cytoplasm viscosity, 2 pN
s/µm^{2}. (B) Trajectories of the centrosome
predicted for the indicated angles between the two synaptic planes. An
excerpt from the experimental trajectory extracted from the
supplementary video to the cited paper ^{2}.

In the light of the model, the pause of the centrosome and of the associated
killing apparatus next to each of the engaged targets appears to arise from the
delayed relaxation of microtubules that were trailing during the last period of
centrosome migration. This can be discerned by close examination of

Simulations in which the pulling force density at the two synapses is unequal
show that the centrosome can be retained at the synapse which is the stronger,
even if it visits the weaker synapse first (

In summary, a purely deterministic, biomechanical model is capable of exhibiting complex, life-like centrosome movements in a conceptually simple, three-dimensional computer simulation of the dynein-pulling mechanism. Our computational results demonstrate that the origin of the strikingly animate wandering of aim in T-killer cells need not be sought necessarily in stochastic dynamics of individual molecules, or in indecision that might be exhibited by complex information processing in the T cell, or in indeterminate changes in the signaling input from the target cells. Instead, the rigorous numerical demonstration that a purely deterministic mechanical explanation exists for one of the most animate behaviors exhibited by cells suggests that similar explanations and supporting experimental evidence can be sought for other types of cell behavior that appear strikingly far from mechanistic.

The T-cell outline in our model is a sphere 14 µm in diameter. It
is truncated by a plane when attachment to the target is modeled. The planar
part of the model cell surface is referred to as the synapse, or synaptic
surface. The entire cell surface is rigid and immobile. The nucleus is also
a rigid sphere (with radius
_{n} = 5
µm).

There are 24 microtubules, each 25 nm in diameter. The microtubule length in
the simulations was 16 µm, except where indicated otherwise.
Effective (hydrodynamic, see below) microtubule diameters between 25 and 50
nm were tried, with similar results. The model microtubules are inextensible
and respond elastically to flexure with the measured rigidity,
^{2}
^{2} were tried, with similar results.) One end of
every microtubule is clamped at the same point on the nuclear surface. This
point is referred to as the centrosome. If unstressed, straight microtubules
would emanate from the centrosome in a uniform conical arrangement
(70° wide unless otherwise specified), but in the model they are
always constrained between the nuclear and the cellular surfaces.
(Unstressed microtubule divergence angles between 60 and 90° were
also tested, with similar results—see

Elastic relaxation of microtubules coupled with the nucleus inside the spherical cell outline comes to a static equilibrium, which is the initial condition for the dynamic simulations. A simulation is begun by intruding the truncating plane at a constant speed into the cell over 25 s to a point where it truncates the sphere by 2 µm. The cell volume is kept constant by a corresponding (minor) increase of the radius of the round part. (Intrusion depths between 1.8 µm and 2.6 µm were tried, with similar results.)

At all times after the beginning of the simulation, microtubules can slide
according to the following rules: When part of a microtubule is within a
small distance (15 nm) from the synaptic surface, force is exerted on that
part of the microtubule. (Contact distances between 10 and 100 nm were also
tried, with qualitatively similar results.) The force is tangential to the
microtubule and directed towards its distal (“plus”)
end. There is a constant magnitude of force exerted per unit length of the
microtubule within the specified contact distance, which is referred to as
the pulling force density. These rules would describe microtubules coming in
contact with the cell cortex on which dynein molecules are anchored at a
certain spatial density, if the dynein is activated upon the synapse
formation as hypothesized

Movement of the microtubules and nucleus is opposed by viscous drag
(overdamped motion, see below). We chose the effective viscosity of the
cytoplasm so as to reproduce the characteristic speed of the centrosome
movements in T cells. To arrive at this value, we proceeded from the drag
coefficient value that was similarly chosen in a comparable type of model
that approximated well the movements during cell division ^{2},
which was used in the simulations shown, turned out to be four times lower.
(Viscosities between 0.1 and 8 pN s/µm^{2} produced
qualitatively similar results.)

As explained above, the length of the each microtubule was kept constant,
with the exception of special additional modeling cases. In these special
additional simulations we tested the impact of dynamic instability
(stochastic changes of microtubule length ^{2}/min. This
value was chosen to lie between those measured in PtK

Microtubules are represented numerically as chains of straight segments that approximate the centerlines of the microtubules. Each microtubule was approximated with 32 segments of equal equilibrium length.

The essential inextensibility of microtubules is implemented by assigning a high Hookean spring constant to the segments. The value of 2000 pN/µm for this constant results in a force restoring the length of the segment which becomes much larger than other typical forces in the model before the segment length change becomes noticeable. In effect, therefore, the microtubules in our numerical model are inextensible and incompressible.

The restoring forces resulting from flexural rigidity of a bent microtubule
were calculated in a slightly more generalized way compared to the
previously developed mitotic spindle model _{i}
_{i}_{i}_{i}
_{i}
_{i−}
_{1}, _{i}
_{i+}
_{1}. There are two co-planar unit vectors, _{i}

To implement our assumption that the microtubule ends are not merely anchored at the centrosome, but clamped there, the above numerical treatment of microtubule bending was applied not only to flexure between actual microtubule segments, but also to the deflection of the first proximal microtubule segment from the direction fixed with respect to the nucleus in the manner described among the physical assumptions.

To implement impenetrability of the cell outline to the microtubules, whenever a node on a microtubule approaches the cell outline closer than the microtubule radius, a reaction force is exerted on that node. The direction of this force is inward normal to the cell outline at the point of its contact with the microtubule. The force magnitude is calculated so that with the drag coefficient associated with the node (explained below), the node will just stop violating the impenetrability condition at the next time step. This definition results in maximally precise implementation of impenetrability without causing numerical instability in our time-stepping scheme. Nucleus violating the cell outline is treated in the same way. Any noticeable penetration of the nuclear volume by microtubules is prevented similarly, with the action and reaction forces similarly calculated and exerted on the nucleus and the microtubule node.

Viscous drag on the nucleus and microtubules was calculated using the same
values of effective viscosity of the cytoplasm, _{n}_{n}^{3}_{n} is the radius of the
nucleus as specified above. The distribution of the drag force along a
microtubule was calculated using the numerical representation of
microtubules as segmented chains, which was described above. The velocity of
a microtubule node _{⊥}) and tangential
(_{∥}) to the microtubule. The drag force
on the node was then calculated as
−2π_{∥}+2_{⊥})

The forward Euler method _{∥}, and orthogonal to it,
_{⊥}. When divided by the appropriate
drag coefficient, these components of force will yield the corresponding
components of the node velocity. Therefore, using the widely accepted
approximations for the drag coefficient of a cylinder (see above), the
position of the node should be updated as follows:

The following procedure is used to update the position and orientation of the
nucleus (together with the centrosome node and the vectors representing the
unstressed microtubule emanation directions, see above). The total force
_{n} applied to the nucleus arises from the
conditions of impenetrability (see above) and from the forces applied to the
centrosome node. The latter are calculated the same way as for the generic
microtubule node (see above). The position of the nucleus center
_{n} is updated as follows, in accordance with the
above formula for drag and with the forward Euler method:

The total force moment is also calculated. It arises only from the forces
applied to the centrosome, at the distance _{n}
from the nucleus center. (All other forces exerted on the nucleus arise from
the impenetrability conditions and are therefore directed to the nucleus
center.) Torque balance with the drag force determines the angular velocity

In those special simulations that incorporated the dynamic instability of
microtubules, the following algorithm was used. At each time step, a
probabilistic decision was made independently for each microtubule, whether
to add a new segment to its free end, remove the end segment, or do nothing.
A pseudorandom number was generated from a uniform distribution between 0
and 1. If the number was smaller than a small parameter ^{2}/Δ

Centrosome reorientation that is caused by cell outline deformation alone, in
the absence of the pulling force. Centrosome orientation is measured as the
angle formed by the vector drawn from the nucleus center to the centrosome
and by the outward normal to the synapse. (I.e. 0 means centrosome pointing
at the synapse and 180°, at the opposite side of the cell.) The thin
straight line is drawn for reference; it indicates where the simulation
results would lie if there would be no reorientation. To generate these
results, the pulling force density in the model was set to zero. Microtubule
length, 16 µm; effective cytoplasm viscosity, 2 pN
s/µm^{2}.

(0.45 MB TIF)

Stabilization of the centrosome next to the stronger synapse. Plotting
conventions are as in ^{2}. The centrosome migration from
the weaker to the stronger synapse appeared irreversible. The large strength
difference was tested because in the experiments that inspired this test,
the antigen load of the target cells differed by a factor of ∼1000

(0.76 MB TIF)

Sensitivity of the model to the value of the unstressed microtubule
divergence angle. (A) Centrosome reorientation plotted for the indicated
values of the unstressed microtubule divergence angle. The ordinate is the
angle formed by the vector drawn from the nucleus center to the centrosome
and the outward normal to the synapse. (The 90° starting angle means
that centrosome in these simulations was initially on the side of the cell
with respect to the synapse.) The plots illustrate relative insensitivity of
the reorientation trajectory to the divergence angle. Pulling force density,
40 pN/µm; microtubule length, 16 µm; effective cytoplasm
viscosity, 2 pN s/µm^{2}. (B) Intra-synaptic oscillations
plotted for the indicated values of the unstressed microtubule divergence
angle. x is the coordinate axis directed across the synapse, as shown in
^{2}.

(0.99 MB TIF)

Reorientation of the centrosome to the synapse. This video corresponds to the
first part of ^{2}.

(3.37 MB MOV)

Reorientation followed by oscillations of the centrosome. Pulling force
density, 20 pN/µm; microtubule length, 16 µm; effective
cytoplasm viscosity, 2 pN s/µm^{2}.

(6.06 MB MOV)

Oscillations in a model which in addition to our usual assumptions
incorporates also dynamic instability of microtubules and a ring-shaped
pulling surface. The area of the synaptic surface where pulling is activated
is shown in black, the parts of the synapse that are inactive as far as
pulling are shown in white. The cell surface is cut out for a clearer view
(only in graphics, not in actual simulation). Pulling force density, 36
pN/µm; starting microtubule length, 16 µm; effective
cytoplasm viscosity, 2 pN s/µm^{2}.

(1.33 MB MOV)

Oscillations in detail. This video is an animation of ^{2}.

(8.61 MB MOV)

Oscillations between two synapses. This video is an animation of the same
simulation that is shown in ^{2}.

(5.51 MB MOV)

We thank Dr. A. Baratt for critically reading the manuscript.