Directed Motility and Dictyostelium Aggregation - Cell Motility - page 80

S 0 ). This condition assures that the overall “balance” between production of
activators and inhibitors is only weakly broken by the decay processes. In this
case, Equation (3.8) simplifies to

k a k − b

k i k −a S 0 = k − b

k i √ K

A 0 ∼ (3.9)

Note that because A 0 ∼ √ S 0 , the signal response A is not perfectly adapting.
However, the transient response can be much larger than the eventual steady
state level. This can be seen in Figure 3.12, where we have plotted the response
to a ten-fold increase in S at t =0 s and again at t =50 s .
When exposed to a shallow gradient, our model is able to create a large
internal asymmetry; furthermore, this internal polarization can be rapidly
reversed if the external information changes (see Figure 3.13). The kinetics
matches the experimental fluorescence data [33, 42]; upon introduction of the
gradient, both the back and the front of the cell respond, followed by a loss
of response at the back. The timescale of the response of A at the back of
the cell, as well as its dynamics upon a gradient reversal, is determined in
large part by the value of the diffusion constant of B . A small value of the
diffusion constant will lead to slow dynamics, while a large value leads to
fast loss of activation at the back and a fast reversal. Figure 3.13 shows that
the timescales in our model can be consistent with experimental findings [42].
Of course, an exact comparison with kinetic experiments requires additional
knowledge of the pathways involved.
Contrary to previous theoretical efforts [43, 39, 44, 41], the response of
the model, measured in terms of the variable A , is not a simple amplification
of the external gradient. Instead, the response should be thought of as a
switch: the front (i.e., the side of the membrane closest to the chemoattractant
source) has a high level of A , while the back has a very low level. For a
wide range of parameters, this leads to an internal asymmetry that is much
larger than the external one (Figure 3.14), as reported in the literature [42].
Moreover, it can be shown [37] that the level at the back is independent of the
external signal for a large range of gradients. Finally, this model is also able
to replicate experiments in which cells are exposed to multiple simultaneous
sources; again, see [37] for a discussion.
What is the mechanism underlying this switch-like and rapidly reversible
asymmetry? The key elements in our model are the equal production of A
and B , together with a suciently large diffusion constant of the cytosolic
component and suciently slow decays. The inclusion of cytosolic diffusion
for B leads to an almost constant concentration of B throughout the cell.
Thus, the initial value of A will be higher than B at the front but lower at the
back. When B jumps onto the membrane as B m , there is more than enough
to eliminate A in the back, but residual A 's will be left over at the front.
The final result is a nearly complete inactivation at the back but not at the
front. And, because this asymmetry is a driven response (i.e., is not created