cap on a disassembling microtubule called res- cue allows the structure to return to the assembly
mode. The dynamic instability phenomenon dis- cussed previously follows from the interplay of
cap disappearance and rescue. In the cell the MTOC adds an additional level of organization
to dynamic instability by anchoring the non- growth end of the microtubules and thereby ori-
enting the growth of the microtubule population as a whole. MAP interactions mold the popula-
tion into different configurations by stabilizing individual microtubules and by subserving a vari-
ety of other organizational roles. The growth dynamics described above provide
four points of regulation. The first is the number of dimers that are converted from straight to
curved in the lateral cap. This regulates the assem- bly characteristics of the individual microtubules.
Second is the number of free dimers that can be incorporated into microtubules. Regulation of the
binding efficiency of structure stabilizing MAPs is the third. The characteristics of the MAPs them-
selves afford the fourth point of regulation. MAPs constitute a large family of proteins. Alterations
in the MAP population can thus influence net- work organization and functionality. These possi-
bilities for regulation suggest that microtubule networks are moldable systems that can develop
into a variety of configurations capable of per- forming different tasks.
3. Model specification
3
.
1
. O6er6iew We picture the microtubule network as embed-
ded in a section of the intra-neuronal environ- ment, represented in terms of a three dimensional
array Fig. 2. Each location in the array is a cube of specified size, indexed in the figure by jkl. The
cube size controls the amount of total protein mass, including MAP as well as microtubule
mass, that can occupy the location. The amount allowed is measured in units of 100 kDa, or the
mass of one tubulin dimer called a dimer unit. One kDa is equivalent to the mass of 1000 hydro-
gen atoms.
Microtubules exclusively occupy one or more contiguous array locations along the l-axis. Each
array location is considered to contain a separate subregion of the microtubule comprising 60
dimers or fewer in the case of the growing tip. This corresponds to a cube edge of 37 nm. The
restriction to one direction of growth results in a population of parallel but potentially interlinked
microtubules similar to natural networks regu- lated by MTOC and MAP organizational effects.
MAPs are represented as independent data structures that diffuse in the array, implemented
Fig. 1. Role of lateral cap in microtubule assembly and disassembly. The conformational state of assembled dimers
may be either straight or curved. Straight dimers favor assem- bly black ovals. Curved dimers favor disassembly white
ovals.
Fig. 2. Microtubule representation. The cell space is repre- sented by a three dimensional array bottom. For computa-
tional convenience each microtubule top is divided into subregions of 60 dimers or fewer at the growing tip that are
placed in this space.
Fig. 3. Cyclic flow of control. The growth dynamics and signal processing modules share a common microtubule network
representation see Fig. 2. The adaptive self-stabilization module couples the growth dynamics to signal processing
performance.
defined training set. Desired outputs are assigned to the input patterns used for training.
The growth module is modified by changing MAP binding affinity and by freezing the micro-
tubule assembly process at a particular learning level at which point learning proceeds only
through MAP diffusion. Increasing the binding affinity increases network stability, while decreas-
ing the affinity decreases stability. When the net- work performs well binding affinity is increased.
The extent of structure change accordingly de- creases. If the network performs poorly binding
affinity decreases, allowing for increased structure change. Freezing the growth process at a certain
learning level protects the network structure from the erratic effects of growth on the signal process-
ing dynamics.
3
.
2
. Growth dynamics The growth module creates a population of
dynamic individuals that exhibit a net stability. The dynamics are based on empirical evidence for
a lateral cap Timasheff, 1991 and computer simulation of cap dynamics in vitro Bayley et al.,
1994. However, our model, as currently formu- lated, does not attempt to match the growth rates
of the natural system.
The following three factors must be taken into account: available straight dimers, dimers bound
in microtubules, and lateral cap size. The pool of available straight dimers at time t, to be denoted
by At, is treated as homogeneously distributed throughout the intracellular environment. The
pool size at time t + 1 given by
At + 1 = At − Gt + d
empty array locations at time t total array locations
1 where Gt is the number of dimers incorporated
in the microtubule network at time t and d is the maximum number of dimers that can be added at
each time step to be referred to as the new dimer coefficient. The ratio of empty array locations to
total locations thus serves as a low-level feedback mechanism ensuring that a consistent mass is
maintained. by stochastically moving the data structure from
one array location to another. MAPs can be in either a bound or free state. Bound MAPs are
part of the microtubule network and influence its dynamics. Accordingly they are not allowed to
diffuse when in this state. Free MAPs are not part of the network, exerting no influence on it, and
thus are allowed to diffuse. Five different MAP types will shortly be introduced. Each MAP type
can consist of subtypes with a unique set of characteristics.
The model consists of three modules forming a cycle Fig. 3, with each iteration representing a
time step t. These modules are: growth dynamics; signal processing; and adaptive self-stabilization.
The growth module uses the array described above to generate a network, doing so in a man-
ner that simulates dynamic instability and that is subject to bound MAP influences. It also imple-
ments diffusion of free MAPs in this array. Signal processing also uses the array, treating each mi-
crotubule as a string of linked discrete oscillators capable of propagating signals that are intro-
duced, manipulated, and extracted by bound MAP activity. The adaptive self-stabilization
module modifies the growth dynamics based on the performance of the signal processing module.
Performance is determined in relation to a pre-
Let g
n
t represent the number of straight dimers incorporated into microtubule n at time t.
This is taken as a random value from 1 to 60. Gt is then the sum of the g
n
t. The size of the lateral cap of microtubule n at time t + 1, to be
denoted by c
n
t + 1, is given by c
n
t + 1 = L
if L ] 0 otherwise
2 where L = c
n
t + g
n
t – r is the lateral cap size at
time t, and r is the number of dimers hydrolyzed, to be referred to as the hydrolysis coefficient.
The above three factors determine the number of array locations a microtubule will occupy, i.e.
how many subregions the microtubule is divided into Fig. 2. For every 60 dimers added the
microtubule will increase by one subregion. This size increase respects array locations currently
occupied by other microtubules and also respects array boundaries. The important point is that
each array location represents a segment of the microtubule, not an individual dimer. This choice
is for computational convenience. Decreasing the number of dimers associated with an array loca-
tion would yield a more refined description; but the computational resources allocated to the rep-
resentation must be balanced against the compu- tational resources required for the learning
process.
If the lateral cap should disappear i.e. if c
n
t = 0 the microtubule will lose one subregion at each time step t unless a bound MAP prevents
disassembly. Rescue,
which switches
micro- tubules from shrinking to growth, is implemented
by randomly generating a value for g
n
t, the number of straight dimers to be incorporated, and
then comparing this to the hydrolysis coefficient r. If the number of added dimers is greater than
the hydrolysis coefficient i.e. the number of dimers lost the lateral cap re-forms and the mi-
crotubule resumes growing. Typical growth behavior is illustrated in Fig. 4.
This shows the percentage of the space occupied by three different microtubule populations. MAPs
were not present in any of these cases. The only difference is in the hydrolysis coefficient r. As
can be seen, the total microtubule mass is in- versely proportional to the hydrolysis coefficient.
The dimer coefficient d has a directly propor- tional effect not shown here.
The stabilizing effect of bound MAPs on the growth dynamics is effectuated by preventing dis-
assembly of the subregion to which they are bound. Unbound or free MAPs are stochasti-
cally rearranged, as would happen in a diffusive process. This is implemented by moving the MAP
one array location in a random direction at each time step t. MAP movement is restricted in two
ways: MAPs cannot leave the array space and they can only move into an array location if the
mass restriction will not be violated.
3
.
3
. Signal processing After the growth dynamics module has com-
pleted its work for a given iteration the signal processing module takes over. Each microtubule
is now treated as a string of coupled oscillators, in the fashion of a loaded string Marion and
Thornton, 1995, except that the time develop- ment is discretized Boole, 1958. The string is
divided into masses connected by springs with each mass representing a microtubule subregion
and the springs representing the collective dimer binding forces. A given mass on the string of
oscillators represents sixty dimers except possibly at the growing tip and also attached MAPs. As
noted in the previous subsection, this lumping increases the computational resources available
for the adaptation experiments. The physical cor- respondence between a discretized loaded string
and the continuous picture implied by lumping is clearly imperfect, since adjacent subregions of the
microtubule are as close to each other as the dimers within these subregions. But for the
Fig. 4. Time variation of a microtubule network size for three different values of r.
Fig. 5. Signal propagation dynamics of a single microtubule containing five subregions. The graph should be read from left
to right, with each time step indicating the state of the microtubule at that time. Thus the displacements from the
equilibrium position represent longitudinal motions in the vertical direction.
MAPs, K is the spring constant, o is a damping coefficient representing energy lost to the environ-
ment, and B
jkl
t is the total displacement due to energy introduced due to active MAP effects. The
boundary conditions are set by eliminating the terms containing x
jkl
+
1
and x
jkl − 1
for the left and right string boundaries, respectively. Thus
we view the string as pinned at both ends. The vibrations are longitudinal if K is taken as
a spring constant, as above. The same equation describes transverse oscillations if K is interpreted
as the tension in the string divided by intermass spacing Marion and Thornton, 1995. The longi-
tudinal picture is more natural here, since the separation of the microtubule into different subre-
gions is rather arbitrary. Fig. 5 illustrates the longitudinal oscillatory behavior of a microtubule
containing five subregions. The parameters used to generate this figure are a total mass M corre-
sponding to 6000 kDa for each subregion, a unit time scale spring constant K of 3000 kDa, and a
unit time scale damping coefficient o of 1000 kDa recognizing that the dimensions are not
biophysically meaningful, due to the discretization of time. The figure shows that the system is at
rest at t = 0, then at t = 1 energy is added to subregion 3 in the form of a mass displacement.
From that point on one can see the wave propa- gating through the structure. Note that nothing
prevents the masses from passing through each other given the potential function used, apart
from an appropriate choice of parameters. Also, a negative value of o can be used to represent
pumping of energy into the network in response to its oscillatory activity, hence a nonspecific form
of amplification.
Adding bound MAPs would have the passive effect of increasing subregion mass and reducing
the oscillatory activity. There are also a number of ways in which MAPs, when triggered, can
introduce energy into the subregion of the micro- tubule to which they are bound, thereby causing
displacement of that subregion. The energy re- lease is here pictured in mechanochemical terms.
Five different MAP types play a role readin, readout, linkers, amplifiers, and modulators.
Readin MAPs are used to introduce a pattern into the network. Readout MAPs extract signals from
present purposes it is sufficient for the signal processing module to provide a variety of wave-
like behaviors that serve to combine signals in space and time in a way that depends on the
structure of the network.
Let x
jkl
t represent the position of the mass associated with array location jkl at time t relative
to its position at rest x
jkl
t = 0. The time t here is distinct from the time t used by the growth
module. This time development is described by x
jkl
t + 1 = 2x
jkl
t − x
jkl
t − 1 +
K M
jkl
[x
jkl + 1
t − 2x
jkl
t +
x
jkl − 1
t] +
o M
jkl
[x
jkl
t − 1 − x
jkl
t] + B
jkl
t 3
where the time increment is set equal to unity. M
jkl
is the total microtubule subregion mass in dimer units at location jkl including all bound
the microtubule network. Linker MAPs are trig- gered by the vibratory activity of a microtubule in
a neighboring array location. Amplifier MAPs respond to the vibratory activity of the location to
which they are bound. Modulators contribute pas- sive effects to the signal process by increasing mass.
The linker and amplifier maps introduce the essen- tial nonlinearities into the network dynamics.
The total mass displacement, B
jkt
t, effected by MAP active effects at location jkl, is the sum of all
displacements produced by readins, linkers, and amplifiers. To release energy into a subregion,
these MAPs must first be triggered. Readin MAPs are automatically triggered to introduce signals
into the network. For linkers and amplifiers the assumption is that they are change detectors that
respond to the change in velocity of a specific subregion. Amplifiers are triggered locally on the
subregion to which they are bound. Linkers are slightly more complex because they are triggered
by a subregion of a neighboring microtubule indi- cated by index j kl. The mass displacement
contributed by a linker, d
jklw
t, is given by d
jklw
t = 9
D
jklw
if a \ T
jklw
otherwise where a = x
j kl
t − 1 − 2x
j kl
t − 2 + x
j kt
t − 3 4
Here D
jklw
is the mass displacement when the MAP w triggers and T
jklw
, is the value at which MAP w at jkl will trigger when compared to the rate of
change at location j kl. This is determined when the MAP binds to the subregion, as is the location
j kl it detects. The direction of the displacement produced by a MAP is randomly assigned at time
of binding. The array location j kl to which it responds is also randomly selected from the 24
possible locations that it could potentially contact.
3
.
4
. Adapti6e self-stabilization Recall that adaptive self-stabilization is essen-
tially error feedback acting on the structure of the microtubule network and on the distribution of
MAPs in a given structure. Picture the microtubule networks as immersed in a broadcast cocktail of
biochemical signals that activate a pattern of readin MAPs in a way that depends on their
specificities. The signals are combined in space and time by the vibratory dynamics of the network.
Readout MAPs may or may not be activated. The error signal depends on the extent to which the
consequent outputs are suitable or unsuitable to the environment. The more the error the greater
the structural variation exhibited by the network. Adaptive self-stabilization can be thought of as a
special case of variation-selection learning, but without a population of reproducing systems for
selection to act on.
As previously noted, the extent of network variation is controlled by the binding affinity of the
MAPs. If the signal processing performance of the network is good the MAPs will adhere to the
microtubule surface strongly, thereby increasing the stability of the network structure. If the perfor-
mance is poor the binding to the surface is weak- ened.
Performance is determined relative to a predeter- mined training set. This consists of a set of input
patterns and a desired output 1 or 0. The training set is processed on time scale t after each growth
cycle iteration on time scale t. The signal process- ing is successful if it produces a 1 output within a
required amount of time for each input pattern that calls for this output and produces no output
during this time period if the input pattern calls for a 0 output. The performance on the training set as
a whole is determined by the percentage of correct responses. The binding affinity increases with this
percentage with increasing slope.
As noted earlier the growth dynamics is largely frozen when a pre-specified level of learning is
achieved. Reaching this level is essentially a ran- dom process, but it ensures that the general struc-
ture is
capable of
supporting minimal
performance above the 50 level. At this point, in the present implementation, we only allow read-
outs, amplifiers, and modulators to alter their positions. Each such MAP can move one array
location, including moves between bound and free states. If these variations lead to an increase in
error then the MAPs move back to their former neighboring positions or back to the former
bound or unbound state. This element of gradient
search is necessary to give direction to the learning process. In an individual-based model as opposed
to one that uses populations variations will have increasingly negative effects as the performance
increases and especially so for a randomly con- structed task such as the one used here. Some
memory mechanism is necessary that allows the system to return to a previous state if the variation
has destructive effects. The fitness function would not otherwise provide direction by selecting better
variations; it would only control the extent to which the microtubules and MAP distribution are
shaken.
4. Learning results
