# Universal catastrophe time distributions of dynamically unstable polymers

Paul B. Dieterle<EMAIL_ADDRESS>Department of Physics, Harvard
University, Cambridge, MA 02138, USA  Jenny Zheng  Department of Molecular and
Cellular Biology, Harvard University, Cambridge, MA 02138, USA  Ethan Garner
Department of Molecular and Cellular Biology, Harvard University, Cambridge,
MA 02138, USA  Ariel Amir  John A. Paulson School of Engineering and Applied
Sciences, Harvard University, Cambridge, MA 02138, USA

###### Abstract

Dynamic instability — the growth, catastrophe, and shrinkage of quasi-one-
dimensional filaments — has been observed in multiple biopolymers. Scientists
have long understood the catastrophic cessation of growth and subsequent
depolymerization as arising from the interplay of hydrolysis and
polymerization at the tip of the polymer. Here, we show that for a broad class
of catastrophe models, the expected catastrophe time distribution is
exponential. We show that the distribution shape is insensitive to noise, but
that depletion of monomers from a finite pool can dramatically change the
distribution shape by reducing the polymerization rate. We derive a form for
this finite-pool catastrophe time distribution and show that finite-pool
effects can be important even when the depletion of monomers does not greatly
alter the polymerization rate.

Phenomena differing in their microscopic details can nonetheless share common
features. The discovery and description of such universal features is one of
the central aims of statistical physics. The identification of universal
behaviors is especially valuable in the modeling of living systems, where many
microscopic details are unknown or difficult to measure. Amazingly, despite
the variety and complexity of biology, some phenomena are observed in many
organisms or multiple two domains of life.

One example is the growth of dynamically unstable polymers, in which a quasi-
one-dimensional polymer elongates, suffers a catastrophic cessation of growth,
depolymerizes, and grows anew. This behavior has been observed in eukaryotic
tubulin and a prokaryotic actin homolog [1, 2, 3, 4, 5].

The pioneering experiments of Mitchison and Kirschner were the first to
demonstrate that the interplay of hydrolysis and polymerization of monomers at
the tip of a growing polymer drives it toward catastrophe [2]. This
observation has been confirmed in several other experiments, also across two
domains of life [1, 6, 7, 3, 8, 4, 9].

Despite much experimental and theoretical work, precise mathematical modeling
of dynamic instability has proven challenging, owing to the difficulty of
probing the microscopic scales of a growing polymer [10, 11]. It is thus
natural to ask which aspects of these models are insensitive to the details.
Here, for a broad class of models, we show that the distributions of times to
catastrophe (henceforth called the “catastrophe time distribution”) are
universally exponential.

The shape of the distribution is insensitive to noise in polymerization and
hydrolysis rates; it is, however, impacted by the depletion of monomers from a
finite pool (which has the effect of lowering the polymerization rate). We
show that this effect can drastically perturb the exponential catastrophe time
distribution, even if it does not greatly alter the polymerization rate. For
several models of dynamic instability, depletion of a finite monomer pool
results in a catastrophe time distribution with a Gumbel-like form.

## Model construction

To model the stochastic growth and hydrolysis of a growing one-dimensional
polymer, we first construct constant-rate models of each process. In such
models, monomers add to the polymer at the tip at a rate, $a$; once added to
the polymer, they hydrolyze at a rate, $b$ (see Figs. 1A and 2A). These
assumptions appear to be experimentally relevant, owing to the fact that
dynamically unstable polymers elongate and hydrolyze at roughly constant rates
[12, 13, 14, 7, 1, 6, 15, 16, 17, 18].

To deduce the catastrophe time distribution for such models, we must first
specify the states of a polymer that result in catastrophic depolymerization.
We will begin by grounding our analysis in two specific models, then deduce
the universal aspects of all such constant-rate models.

Figure 1: Cap model dynamics. A: Schematic of the cap model. Polymerization
and hydrolysis proceed at constant rates – $a$ and $b$, respectively. The
polymer has a cap region of size $n$. When $m$ of the cap region monomers
hydrolyze, the polymer catastrophically depolymerizes. Cap models can be
described by a Markov Process with transition matrix $\mathbf{M}$. B:
Eigenspectrum of the cap model transition matrix, $\mathbf{M}$, shown here for
various cap sizes and $b/a=0.1$. In all cases, the eigenspectrum shows a
quiescent eigenvalue with small negative real component (red dot). This small
negative real eigenvalue dominates the catastrophe dynamics. There is an
eigenvalue at zero which is not shown here. C: Catastrophe time distribution
for the $n,m=2$ cap model with $b/a=0.1$. A very short transient (which
depends on the initial conditions) is followed by an exponential distribution
described by a characteristic timescale, $T$. D: Characteristic timescale,
$T$, as a function of $(b/a)^{m}$ for various cap sizes. In the limit of
$nb/a\ll 1$, the timescale for all models goes as $aT\sim(b/a)^{m}$.

### Cap model dynamics

First, we consider so-called cap models, in which catastrophe is caused by the
hydrolysis of at least $m$ out of the $n$ monomers nearest the growing (see
Fig. 1). We refer to these $n$ monomers the “cap region”. Once the polymer
experiences a catastrophe, it irrevocably depolymerizes back to length zero.
(We note that microtubules do not strictly display the latter behavior, as
they occasionally arrest their depolymerization and regrow before reaching
zero length. However, complete depolymerization has been observed in bacterial
actin homologs. By neglecting “recovery and regrowth”, we are in effect
studying a previously reported “bounded growth” phase [19, 20].) Such cap
models have been explored in several previous works, which have calculated
some of the model dynamics when $n=m$ [14, 21].

To discover the catastrophe time distribution in such models, we begin by
enumerating the allowed states of the cap of the polymer (Fig. 1A). An
individual monomer is either hydrolyzed () or unhydrolyzed (), so that, e.g.,
the state space for $n,m=2$ is $\\{\leavevmode\hbox to4.71pt{\vbox
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constant rate models we consider here, the state  transitions to the states
or  by hydrolyzing at rate $b$; the state  transitions to  by polymerizing at
rate $a$. The flow between these states is thus governed by a transition
matrix, $\mathbf{M}$, so that the probability of being in each state,
$\mathbf{p}=(p_{\leavevmode\hbox to3.41pt{\vbox
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dynamical equation

$\frac{d\mathbf{p}}{dt}=\mathbf{M}\mathbf{p}\implies\mathbf{p}(t)=e^{\mathbf{M}t}\mathbf{p}_{0}$
(1)

with $\mathbf{p}_{0}$ the initial probability state vector. Henceforth, we
will refer to the last entry in the state vector as the absorbing state (with
$m$ out of $n$ monomers hydrolyzed) by index $N$.

To find the catastrophe time distribution, $p_{c}(t)$, we may find the
transition rate into the absorbing $N$th state (in which $m$ out of $n$ of the
cap monomers are hydrolyzed):

$p_{c}(t)=(\mathbf{M}e^{\mathbf{M}t}\mathbf{p}_{0})_{N}.$ (2)

By numerically solving eq. (2) we observe (see Fig. 1C) that for polymers that
grow to lengths $l\gg 1$, $p_{c}(t)$ is well-described by an exponential
distribution with timescale $T\gg 1/a,1/b$ (plus some transient behavior at
times $t\approx 1/b$ that depends on the initial condition). This is true
except in the case of $n,m=1$, for which the distribution is exactly
exponential with timescale $1/b$, corresponding to the waiting time of a
single Poisson event – namely, the hydrolysis of the monomer at the tip.

To understand the nature of this nearly exponential distribution, we
characterize the eigenspectrum of the transition matrix, $\mathbf{M}$. We
observe (see Fig. 1B) a constellation of eigenvalues in the complex plane. The
negative real components of all but one of these eigenvalues are far from
zero; it is this solitary small negative real eigenvalue that sets the
timescale of the exponential distribution that dominates the catastrophe time
distribution.

By plotting the catastrophe timescale, $T$, as a function of $(b/a)^{m}$, we
see (Fig. 1D) that

$b/ma\ll 1\implies
T\approx\frac{A_{\text{cap}}}{a}\left(\frac{a}{b}\right)^{m},$ (3)

where $A_{\text{cap}}$ is an $n,m$-dependent constant (see Table 1). This
relationship has been previously noted by Brun et al. for the case of $n=m$
[14]. Here, we provide some intuition for how it comes about in general.
Consider a process by which a monomer is added to the filament in every time
interval of length $1/a$. At each time step, we draw a random configuration of
the $n$ monomers in the cap; the monomer $k$ away from the tip has thus been
in the filament for a time $k/a$; correspondingly, the probability that it has
been hydrolyzed since its addition is $p_{h}(k)=1-e^{-kb/a}$. If we focus on
$k\leq n$ with $nb/a\ll 1$, then $p_{h}(k)\approx kb/a$. The probability of
encountering $m$ hydrolyzed monomers in the cap region thus scales as
$(b/a)^{m}$ and the mean catastrophe time scales as in eq. (3). This intuition
is not strictly rigorous because it treats the polymer state at each time step
as independent from previous time steps. However, the state of the cap is
completely erased after $n$ polymerization events and thus the assumption of
independence should be accurate when the mean catastrophe time, $T$, greatly
exceeds $na$.

Figure 2: Growth-and-shrinkage model dynamics. A: Schematic of the growth-and-shrinkage model. Polymerization and hydrolysis proceed at constant rates of $a$ and $b$, respectively. If $m$ out of the first $n$ monomers become hydrolyzed, the filament becomes unstable and depolymerizes until it either finds a stable state (at which point it arrests depolymerization and regrows) or catastrophically depolymerizes. B: Numerically simulated histogram of 5000 catastrophe events for $n,m=2$ and $a/b=5$. As with cap models, growth-and-shrinkage models are governed by an exponential catastrophe time distribution. C: Numerically simulated elongation rate, $r/b$, as a function of the normalized polymerization rate, $a/b$. For the small $n,m$ growth-and-shrinkage models we tested, the effective elongation rate goes as $r/b=a/b-x$. The black line of $r/b=a/b$ is a guide to the eye. D: Numerically simulated catastrophe time, $bT$, as a function of the normalized polymerization rate $a/b$. Colors are the same as in panel C. Dashed black lines are exponential fits. The solid black line is the calculated mean catastrophe time using Markov chains, (5). The mean catastrophe time grows exponentially in $a/b$ for every small $n,m$ model we tested. $(n,m)$ | $(1,1)$ | $(2,1)$ | $(2,2)$ | $(3,1)$ | $(3,2)$ | $(3,3)$
---|---|---|---|---|---|---
$A_{\text{cap}}$ | 1 | 0.5 | 0.33 | 0.33 | 0.086 | 0.079
$A_{gs}$ | 1.59 | 1.86 | 1.61 | 2.52 | 1.50 | 1.54
$s$ | 0.96 | 0.37 | 0.96 | 0.19 | 0.48 | 0.96
$x$ | 1.20 | 3.54 | 0.53 | 0.81 | 1.33 | 0.30
Table 1: Model parameters. Extracted model parameters of $A_{\text{cap}}$ for
the cap model and $A_{gs}$, $s$, and $x$ for the growth-and-shrinkage model.

### Growth-and-shrinkage model dynamics

As appealing as cap models are for their conceptual simplicity, they fail to
explain why a depolymerizing filament that encounters a stable cap region
(say, , in the case of $n,m=2$) does not arrest its depolymerization and
regrow. To that end, Antal et al. have proposed a different class of growth-
and-shrinkage models, in which a polymer tip may transition to a catastrophe
state, depolymerize until it finds a stable cap region within the bulk of the
polymer, arrest its depolymerization, and regrow (Fig. 2A) [22, 23].

We assume that a filament is stable when a cap region of length $n$ has at
most $m-1$ hydrolyzed monomers within it. In this state, the filament
polymerizes at rate $a$. Each monomer in the filament hydrolyzes irreversibly
at rate $b$. And when $m$ of the cap monomers hydrolyze, the polymer is
destabilized. This instability causes depolymerization, which ceases only if
the polymer depolymerizes into a stable cap region; if it fails to find such a
stable region, it catastrophically shrinks to zero length. We assume, for
mathematical convenience and in line with experiments in microtubules and
actin homologs, that the depolymerization rate greatly exceeds the
polymerization rate, and that the depolymerization process can thus be
regarded as instantaneous.

This cycle of constant growth and shrinkage changes the dynamics of the
polymer as compared to the cap model. First of all, the apparent elongation
rate of the polymer is no longer the polymerization rate, $a$, but some
complex function $r(a,b,n,m)$. Secondly, the mean catastrophe time (the mean
time from the beginning of growth to catastrophic depolymerization) is
dramatically extended, owing to the fact that a depolymerizing filament can
arrest the process by finding a stable cap region buried in the bulk.

To characterize the catastrophe time distribution, the mean catastrophe time,
and the effective elongation rate, $r$, we performed simulations of the
polymer growth and catastrophe process for various $n,m$ using the Gillespie
Algorithm. Our findings are summarized in Figure 2. We show that the
catastrophe time distribution of growth-and-shrinkage models also follows an
exponential distribution, that the mean catastrophe time grows exponentially
as $aT=A_{gs}\exp(sa/b)$ with $s$ and $A_{gs}$ both $n,m$-dependent constants
(see Fig. 2 and eq. (5)), and that $r\approx a-bx$ with $x$ an $n,m$-dependent
constant. See Table 1 for various values of $A_{gs}$, $s$, and $x$. All of
these dependencies deviate strongly from the corresponding dynamics within the
cap model.

We note that the mean catastrophe time of all models with $n=m$ is the same;
this is because the catastrophe condition within these models is the same:
every monomer must be hydrolyzed. We can calculate the relevant timescale by
considering the probability of the polymer having $j$ unhydrolyzed monomers at
a given time $t$, $p_{j}(t)$. States with $j$ unhydrolyzed monomers can make
one of two transitions: (1) a monomer hydrolyzes and the polymer transitions
into a state with $j-1$ unhydrolyzed monomers (this happens at rate $jb$) or
(2) the polymer transitions to a state with $j+1$ unhydrolyzed monomers by
polymerizing (at rate $a$). Thus, the dynamics are given by:

$dp_{0}/dt=bp_{1}$ (4a)

$dp_{1}/dt=2bp_{2}-(a+b)p_{1}$ (4b)

$j\geq 2:~{}dp_{j}/dt=(j+1)bp_{j+1}+ap_{j-1}-(a+jb)p_{j}$ (4c)

These dynamics again yield a transition matrix, $\mathbf{M}$, with a unique
small negative real eigenvalue whose inverse is the exact mean catastrophe
time. We can analytically calculate an approximate timescale by considering
the mean time to the absorbing ($j=0$) state when starting from a singly
unhydrolyzed ($j=1$) state. This timescale, $T_{1}$, can be calculated using
properties of Markov chains [24] and is given by

$aT_{1}=\sum_{k=1}^{\infty}\prod_{j=1}^{k}\frac{a}{jb}=\sum_{k=1}^{\infty}\frac{(a/b)^{k}}{k!}=e^{a/b}-1\approx
e^{a/b},$ (5)

which closely tracks the results from numerical simulation (see Fig. 2D) and
the fit values of $s$ in Table 1.

Antal et al. have previously calculated the mean catastrophe time and
polymerization rate for the $n=m=1$ growth-and-shrinkage model by considering
the probability that every monomer ever added to a given filament becomes
hydrolyzed (a necessary condition for complete depolymerization when $n=m=1$)
[22]. Their relationship, $aT\approx\sqrt{\frac{a}{2\pi b}}e^{\pi^{2}a/6b}$,
is roughly exponential in its functional form but treats the polymer state at
each time step as independent from the previous time steps and thus
overestimates the mean catastrophe time.

### The universal distribution of constant-rate models

The two models we have considered so far differ greatly in their mean
catastrophe times and effective elongation rates. They rely on different
underlying assumptions about the conditions that cause catastrophe. And yet,
they display the same shape of the catastrophe time distribution. Why?

As we will now show, the shape of the catastrophe time distribution for models
with constant polymerization and hydrolysis rates is universally exponential.
Thus, the precise catastrophe condition is irrelevant; we ought always to
expect an exponential distribution.

The reason for this universal behavior can be gleaned by considering a polymer
that adds a monomer every $1/a$ time interval and has some catastrophe
condition within a cap of size $n$. Suppose we happen upon the polymer and see
it growing, but do not know its state. It has some probability $p_{c}$ of
undergoing catastrophe prior to the next polymerization event. After $n$
polymerization events, the initial state of the cap is totally erased and its
catastrophe probability prior to the next polymerization event is still
$p_{c}$; the same thing is true after $n+1$ polymerization events. Thus, the
catastrophe time distribution of the filament is simply the waiting time
distribution of a single Poisson event, and is thus exponential for times
$T>n/a$. (The condition $T>n/a$ is necessary to erase the polymer’s “memory”
of the initial condition.) That this phenomenon is universal within a broad
class of models justifies previous phenomenological theories, which assumed a
constant catastrophe rate [19, 25, 26].

In Appendix A, we show that this prediction is robust even to noise in the
ratio of $a/b$, so that experiments that construct a catastrophe time
distribution by binning events in slightly varying experimental conditions
should still observe a roughly exponential distribution.

Figure 3: Finite pool model dynamics. A: Dynamics of finite pool cap models.
Left: Catastrophe time distributions with changing pool size, $N_{0}$. The
histograms are numerical simulations of 5000 catastrophe events and the black
line is eq. (11). The theory is expected to hold when $nb/a\ll 1$, which is
roughly the case here ($n,m=3$, $b/a_{0}\approx 0.043$). For pool sizes much
larger than $a_{0}/R_{0}$, the catastrophe time distribution is approximately
exponential. As the pool size shrinks, the catastrophe time distribution
becomes distinctly non-exponential. Right: Kymograph showing the filament
length as a function of time with an initial pool size of $N_{0}=1500$. In
these conditions, we can see a slowing down of the polymerization rate along
with a change to the catastrophe time distribution. B: Dynamics of finite pool
growth-and-shrinkage models. Right: Catastrophe time distributions with
changing pool size, $N_{0}$. The histograms are numerical simulations of 5000
catastrophe events; the solid black line is the nearly exact distribution, eq.
(15); and the dashed black line is the Gumbel-like approximate distribution,
eq. (16). Here, we omit catastrophe events at times $t<b/a_{0}$ as these
depend sensitively on the initial condition of the polymer. Again, we observe
distinctly non-exponential catastrophe time distributions as the pool size,
$N_{0}$, shrinks and becomes comparable with $a_{0}/R_{0}$. Left: Kymograph
showing the filament length as a function of time with an initial pool size of
$N_{0}=1000$. The deviation from an exponential distribution is large, despite
little apparent slowdown in the polymerization rate: in the conditions of the
plot, a polymer of 250 monomers will polymerize at 75% of the initial rate and
undergo catastrophes at approximately 3.5 times the initial rate.

## Finite pool models

So far we have treated the polymerization rate, $a$, as a constant. This seems
a natural assumption given the roughly constant polymerization rate seen in
experiments. However, as we have seen, the catastrophe timescale within
certain models can be very sensitive to the ratio of $b/a$. Thus, it is
natural to ask how the catastrophe time distribution is altered when monomers
are depleted from a finite pool, thus lowering the polymerization rate. We
should expect – and will shortly demonstrate – that alterations to the
catastrophe distribution will be substantial when the initial pool size of $N$
monomers is roughly of order the mean catastrophe time of a polymer. Several
previous works have also used numerical simulation and analytics to show
qualitative changes to catastrophe dynamics within a different set of
phenomenological models [27, 28].

Here, we will treat polymerization as a first-order process in which the
polymerization rate, $a$, is proportional to the local concentration of
monomers in the pool. This is valid in the limit that monomers attach by
simply “running into” the tip of the growing polymer, which is thought to be
the case for tubulin. This assumption appears to hold for microtubules, which
show a linear dependence of elongation rate on tubulin concentration [16]. We
will further assume that the finite pool is well-mixed, so that the
polymerization rate is proportional to the total number of monomers in the
pool, $N$. Protein diffusion constants are of order $D\sim 10^{-12}-10^{-11}$
m2/s [29], so that for a typical polymerization rate of order $a\sim 1$/s, a
monomer can diffuse a length $l\sim\sqrt{D/a}\approx 1~{}\mu$m of order the
size of the cell; we thus expect the well-mixed assumption to hold. Finally,
we will assume that the monomer pool is relatively large, so that each
polymerization event only slightly changes the pool size.

We will first work out the general dynamics of such systems, then we will
probe the specific dynamics for both the cap model and the growth-and-
shrinkage model.

Once we specify a polymerization rate, a hydrolysis rate, and the polymer
states that result in catastrophe, we can calculate the rate, $R(a,b)$, at
which the polymer experiences catastrophes, which for the models we have
studied here is given by the inverse of the mean catastrophe time:
$R(a,b)=1/T(a,b)$. The catastrophe rate acquires a time dependence through the
ever-changing polymerization rate, $a(t)$. The probability that the filament
survives without catastrophe until time $t$, $P_{s}(t)$, is thus governed by
the following differential equation:

$\frac{dP_{s}}{dt}=-R(a(t),b)P_{s}.$ (6)

By formally integrating this equation and taking its time derivative, we can
construct the catastrophe time distribution, $p_{c}(t)=-dP_{s}/dt$:

$p_{c}(t)=R(a(t),b)\exp\left[-\int_{0}^{t}d\tilde{t}~{}R(a(\tilde{t}),b)\right].$
(7)

To find $p_{c}(t)$, we must therefore specify a catastrophe condition, which
will determine the catastrophe rate, $R$. We must also determine the time
dependence of the polymerization rate, $a$. Because $a$ is proportional to the
monomer pool size, $N$, it suffices to specify an equation for $dN/dt$ – how
the pool size depletes over time. Both of these calculations are tractable
within the models we have studied so far.

### Finite-pool cap models

Let’s assume that the initial pool size is $N_{0}$ and that the initial
polymerization rate is $a_{0}$. Thereafter, the polymerization rate $a$ will
be given by $a(t)=a_{0}N(t)/N_{0}$. Within the cap model, the number of free
monomers in solution decreases as

$\frac{dN}{dt}=-a(t)=-a_{0}N(t)/N_{0}\implies N(t)=N_{0}e^{-a_{0}t/N_{0}}.$
(8)

It follows that the polymerization rate, $a(t)$, is given by

$a(t)=a_{0}N(t)/N_{0}=a_{0}e^{-a_{0}t/N_{0}}.$ (9)

We have previously seen that when $a/b\gg n$ for cap models with cap size,
$n$, and critical damage number, $m$, the catastrophe rate is given by
$R=1/T=a(b/a)^{m}/A_{\text{cap}}$ with $A_{\text{cap}}$ a constant depending
on $(n,m)$ (see Table 1). Thus, setting
$R_{0}=a_{0}(b/a_{0})^{m}/A_{\text{cap}}$ and $R_{1}=(m-1)a_{0}/N_{0}$, we
have

$R(t)=R_{0}e^{R_{1}t}.$ (10)

We can therefore see that the catastrophe rate grows exponentially for cap
models of $m>1$ within this limit. We can determine the catastrophe time
distribution directly by plugging eq. (10) into eq. (7):

$p(t)=R_{0}e^{R_{0}/R_{1}}e^{R_{1}t-\frac{R_{0}}{R_{1}}e^{R_{1}t}}.$ (11)

As we will see, this distribution (which takes on a Gumbel-like form) also
governs the dynamics of some growth-and-shrinkage models. We emphasize that
for cap models, it only applies when $a/b\ll n$; however, it captures some
qualitative features of the actual catastrophe time distribution even when the
agreement is not perfect (see Fig. 3A).

We have already noted that cap models have a catastrophe rate, $R$, that
depends on the polymerization rate as $R\sim a^{1-m}$. The polymerization rate
depends linearly on the pool size, $N$, so that $R\sim N^{1-m}$. For large
$m$, $R$ can therefore depend very sensitively on $N$ – much more sensitively
than the linear dependence of $a\sim N$ – and we can have large changes to the
catastrophe time distribution with relatively small changes to the
polymerization rate (see Fig. 3).

### Finite-pool growth-and-shrinkage models

The processes underlying growth-and-shrinkage models deplete monomers from the
pool at the effective elongation rate, $r$. As noted previously, this rate can
differ from the polymerization rate, $a$, and is well-approximated by $r=a-bx$
with $x$ set by the cap parameters, $n$ and $m$ (see Table 1 and Fig. 2).
Thus, we have that

$\frac{dN}{dt}=-r(t)=-a(t)+bx\\\ \implies
N(t)=\frac{N_{0}}{a_{0}}\left[bx+(a_{0}-bx)e^{-a_{0}t/N_{0}}\right]$ (12)

while

$a(t)=a_{0}N(t)/N_{0}=bx+(a_{0}-bx)e^{-a_{0}t/N_{0}}.$ (13)

From our analysis of growth-and-shrinkage models we know that
$A_{gs}R=ae^{-sa/b}$ with $A_{gs}$ and $s$ both $n,m$-dependent constants (see
Table 1). Because $R$ depends so sensitively on $a/b$, a small decrease in
$a/b$ due to depletion of the finite pool will cause a large increase in $R$
and quick catastrophe. We may therefore derive a nearly exact catastrophe time
distribution by writing $a/b\approx a_{0}/b+\delta$ with $\delta\ll a_{0}/b$,
which allows us to conclude that

$A_{gs}R=ae^{-sa/b}=be^{-sa/b+\log\frac{a}{b}}\approx\\\
a_{0}e^{-\frac{a}{b}(s-b/a_{0})-1}.$ (14)

Setting $\tilde{s}=s-b/a_{0}$, $\tilde{A}=eA_{gs}b/a_{0}$,
$R_{i}=be^{-\tilde{s}x}/\tilde{A}$, $R_{1}=a_{0}/N_{0}$,
$y=\frac{\tilde{s}}{b}(a_{0}-bx)$, and $R_{0}=R_{i}e^{-y}$ and following the
mechanics of the previous section yields the catastrophe time distribution

$p_{c}(t)\approx R_{i}~{}\exp\bigg{\\{}-ye^{-R_{1}t}\\\
-\frac{R_{i}}{R_{1}}\left[\text{Ei}(-y)-\text{Ei}(-ye^{-R_{1}t})\right]\bigg{\\}},$
(15)

with $\text{Ei}(.)$ the exponential integral function. We interpret $R_{0}$ as
the initial catastrophe rate of the growing filament and $R_{1}$ as a
reparameterized initial pool size.

At first glance, eqs. (11) and (15) appear unrelated. However, when the
initial polymerization rate divided by the initial catastrophe rate is much
less than the initial pool size, we have $R_{0}\gg R_{1}$. In this limit,
catastrophes occur overwhelmingly at times $t\sim 1/R_{0}\ll 1/R_{1}$ and the
catastrophe rate simplifies to $R(t)\approx R_{0}e^{yR_{1}t}$ so that the
catastrophe time distribution is given by

$R_{0}\gg R_{1}:~{}p_{c}(t)\approx
R_{0}e^{R_{0}/yR_{1}}e^{yR_{1}t-\frac{R_{0}}{yR_{1}}e^{yR_{1}t}},$ (16)

exactly the same form as eq. (11). Thus, within certain limits, the finite-
pool distributions of these models obey a common Gumbel-like shape. We explore
the dynamics of the finite-pool catastrophe time distribution in Figure 3B,
where we benchmark the dynamics of eqs. (15) and (16) against simulations. As
with finite-pool cap models, we observe that the removal of monomers from a
finite pool can dramatically affect the catastrophe time distribution even
when the polymerization rate is relatively unchanged. Actually, the effect is
even stronger than for cap models, owing to the fact that while the
polymerization rate goes like the pool size, the catastrophe rate is
exponential in the polymerization rate (and hence the pool size).

## Discussion

In this work, we have elucidated universal and common features within models
of dynamic instability. Such universal features are important for studies of
biophysical phenomenon, as they offer predictions that are largely independent
of microscopic details.

Experiments probing the catastrophe time distributions of microtubules have
discovered that dynamic instability on the minus end is governed by an
exponential distribution while the plus end is distinctly non-exponential [6].
This discrepancy is interesting from a modeling perspective as it supplies an
opportunity to investigate the interplay of structure and dynamic instability.
Odde et al. show that the observed non-exponential distribution can be
explained by interaction between the quasi-one-dimensional microtubule
protofilaments, each of which obeys an exponential catastrophe time
distribution. The latter assumption also underlies well-known phenomenological
models of dynamic instability [19, 25, 26]. We have shown that such an
assumption is valid within a broad class of models.

Other experimental observations – including the adjustment of catastrophe
times post-dilution – have also been understood as arising from interactions
between the one-dimensional protofilaments that comprise the microtubule [21,
14]. Moreover, these interactions can be complicated by mechanical and
chemical couplings [10, 11, 30, 31]. It remains to be seen which of the
features we have unearthed survive in these more complex settings. It would
also be interesting to investigate theoretically the nucleation properties of
dynamically unstable polymers [32, 33].

Our work here focuses on dynamic instability of isolated one-dimensional
polymers. It would be interesting to study the catastrophe and finite-pool
dynamics of many dynamically unstable polymers growing simultaneously. Such
studies could be compared with related studies of polymers that are not
dynamically unstable [34, 35].

The complex geometrical structure of the microtubule makes it a difficult
object to model. Direct application of our models may be more plausible for
quasi-one-dimensional actin homologs [1, 36]. By cultivating a deep
understanding – bridging theory and experiment – of these simple polymers, we
may begin to understand the precise details of the universal phenomena we have
detailed here.

## Acknowledgements

PBD acknowledges support through the Paul M. Young Fellowship of the Fannie
and John Hertz Foundation. JZ acknowledges support through The NSF-Simons
Center for Mathematical and Statistical Analysis of Biology at Harvard (award
#1764269) and the Harvard Quantitative Biology Initiative.

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## Appendix A: Sensitivity analysis of the exponential distribution

Figure 4: Sensitivity analysis of the exponential distribution. A: Catastrophe
time distribution – on linear (top) and semi-log (bottom) scales – of a cap
model with inter-experimental variation in the polymerization rate, $a$. We
assume the polymerization rate is drawn from a gamma distribution with mean
value, $a$, and standard deviation $\sigma_{a}=0.1a$. Even with added noise,
the catastrophe time distribution (solid gray line) is almost exactly the same
as the zero-variability distribution (dashed black line). B: Same as A but
with the growth-and-shrinkage model. The distribution with noisy $a$ is given
by the solid blue line while the dashed black line is the distribution with
fixed $a$.

Here, we show that our main finding – the ubiquity of the exponential
catastrophe time distribution within constant-rate models – is insensitive to
noise. Imagine that you would like to observe the exponential catastrophe time
distribution, but that the polymerization rate, $a$, changes between
experiments. We will show that the exponential catastrophe time distribution
is robust to such noise.

To do so, we assume that the polymerization rate in each experiment is drawn
at random from some distribution, $q(a^{\prime})$ with mean $a$ and standard
deviation $\sigma_{a}$. With fixed $b$, the mean catastrophe time becomes a
function of the particular value of $a^{\prime}$ drawn from the distribution.
Each $a^{\prime}$ will correspond to an exponential distribution with mean
catastrophe time $T(a^{\prime},b)$, so that the empirical catastrophe time
distribution will be given by

$p_{c}(t)=\int_{0}^{\infty}da^{\prime}~{}q(a^{\prime})\frac{e^{-t/T(a^{\prime},b)}}{T(a^{\prime},b)}.$
(17)

To check whether $p_{c}(t)$ is indeed well approximated by the noise-free
distribution $e^{-t/T(a,b)}/T(a,b)$, we must make some assumptions about
$q(a^{\prime})$. For mathematical convenience, we pick $q(a^{\prime})$ as a
gamma distribution with mean $a=k\theta$ and variance
$\sigma_{a}^{2}=k\theta^{2}$. In Figure 3B, we show that under these
assumptions, the noisy catastrophe time distribution differs only very
slightly from the noise-free distribution for noise at the level of
$\sigma_{a}/a=0.1$.


tableTesi [tableTesi] width = colspec =X[1]X[1]X[1], columns =l, hlines,
vlines, row1 = bg=bluepoli, fg=white, font=, c,

# A Gamified Framework to Assist Therapists with the ABA Therapy for Autism
††thanks: Contact authors: Matteo Cordioli (matteo.cordioli@mail.polimi.it),
Dr. Laura Delfino (laura.delfino@solcomantova.it), Dr. Alessia Romani
(aleromani24@gmail.com), and Dr. Elisa Mortini (elisa.mortini1@gmail.com).
Pier Luca Lanzi<EMAIL_ADDRESS>0000-0002-1933-7717).

Matteo Cordioli2, Laura Delfino1, Alessia Romani1, Elisa Mortini1, Pier Luca
Lanzi2

1Spazio Autismo Cooperativa Sinergie — Mantova, Italy
https://www.solcomantova.it/servizi/autismo/ 2Dipartimento di Elettronica,
Informazione e Bioingegneria (DEIB) — Politecnico di Milano, Milan, Italy

###### Abstract

We present a framework to assist therapists and children with autism spectrum
disorder in their Applied Behavioral Analysis (ABA) therapy. The framework was
designed in collaboration with Spazio Autismo, an autism center in Mantova,
Italy. The framework is a first step toward transitioning from the current
paper-based to fully digital-supported therapy. We evaluated the framework
over four months with 18 children diagnosed with classic autism, ranging from
4 to 7 years old. The framework integrates a mobile app that children and
therapists use during the sessions with a backend for managing therapy
workflow and monitoring progress. Our preliminary results show that the
framework can improve the efficacy of the therapy sessions, reducing non-
therapeutic time, increasing patient focus, and quickening the completion of
the assigned objectives. It can also support therapists in preparing learning
materials, data acquisition, and reporting. Finally, the framework
demonstrated improved privacy and security of patients’ data while maintaining
reliability.

###### Index Terms:

Autism, Applied Behavioural Analysis, Serious Games, Gamification

##  I Introduction

Autism is a neurodevelopmental disorder characterized by challenges in social
interaction, communication difficulties, and repetitive patterns of behavior
or interests. It is usually diagnosed in early childhood, and its exact cause
is still not fully understood. A cure for autism does not currently exist;
however, behavioral, developmental, educational, and psychological
intervention strategies can significantly improve the lives of individuals [1,
2].

In recent years, serious games have shown to be valuable potential as tools
for aiding the treatment of individuals with autism [3, 4]. This paper
presents a gamified framework we developed to assist therapists and children
with autism spectrum disorder during therapy. It has been designed in
collaboration with Spazio Autismo in Mantova, a center specialized in
supporting individuals of all ages with ASD. Its mission is to enhance the
quality of life for individuals with autism and their families by promoting a
cohesive and integrated approach. This is achieved through collaborative
efforts, with a strong focus on working with families as the initial step
towards creating a meaningful life plan.

Spazio Autismo is staffed with a team of psychologists, consultants, and
educators who act as case managers for the families of individuals with
autism. The center applies the Applied Behavior Analysis (ABA) therapy [5, 6],
one of the most effective behavioral therapies that utilizes principles of
learning theory to generate positive behavior changes and improve the quality
of life for individuals with autism.

Our framework has been conceived as a first step toward transitioning from the
paper-based protocol currently followed at the center to a fully digital-
supported therapy. It has been designed to align faithfully with ABA
guidelines, support the entire flow of therapeutic sessions, help therapists
create personalized sessions, and prepare its learning material. The framework
comprises (i) an app for tablets for children to use during the sessions; (ii)
a backend that records performance and behavior information; and (iii) a web
frontend for therapists to plan sessions and review patients’ progress.

We evaluated our framework over four months with 18 children diagnosed with
classic autism, ranging from 4 to 7 years old. Our preliminary results show
that time spent on non-therapeutic activities has been reduced with an
increase between 10% to 68% of patients’ focus. They also show a significant
increase in the number of learning objectives completed compared to the paper-
based protocol over the same period. In addition, the framework helped
therapists collect more information about the sessions and patients’ progress,
reduce the time needed to prepare the learning material, and share the results
with the families by automatically generating progress reports.

##  II Autism and Applied Behavioral Analysis (ABA)

Autism encompasses a spectrum of disorders, each with its unique
characteristics [7], and it manifests differently in each individual [8].
There are three main types of autism: Autistic Disorder, Asperger’s syndrome,
and pervasive developmental disorder not otherwise specified (PDD-NOS).
Autistic Disorder, or classic autism, is characterized by significant
impairments in communication, social interaction, and the presence of
repetitive behaviors. Individuals with classic autism may exhibit delayed
language development, difficulty forming relationships, repetitive movements,
or fixated interests. This type of autism is often associated with
intellectual disability. Asperger’s Syndrome is considered a milder form of
autism. Individuals with Asperger’s may have average to above-average
intelligence but struggle with social interaction and communication. PDD-NOS
is a diagnosis often used for individuals with milder symptoms that do not
meet the criteria for either classic autism or Asperger’s syndrome.

###  II-A Applied Behavior Analysis (ABA)

Applied Behavior Analysis (ABA) [5, 6] is one of the most effective behavioral
therapies that utilizes principles of learning theory to generate positive
behavior changes and improve the quality of life for individuals with autism
and other developmental disorders. It is a data-driven, evidence-based
approach that aims to bring meaningful and lasting behavior changes in
individuals. By applying scientifically validated strategies, ABA helps
individuals with autism and other developmental disorders to improve their
skills, enhance their independence, and lead fulfilling lives [9].

ABA involves systematic assessment and intervention strategies to target
socially significant behaviors. It begins with a comprehensive evaluation to
identify the individual’s strengths, challenges, and specific behaviors that
must be addressed. Based on the initial assessment, ABA interventions are
tailored to each person’s unique needs and abilities. Data collection and
analysis are integral components of ABA, as they provide objective information
about behavior patterns, progress, and the effectiveness of interventions.

###  II-B ABA Therapy Design

The frequency of ABA sessions can vary depending on several factors, including
the individual’s needs, goals, available resources, and unique circumstances
of each individual [10]. ABA is considered a long-term intervention rather
than a short-term fix. Consistency and continuity of therapy are important for
skill development and behavior change. For some individuals, ABA may be needed
for several years to address a range of skills and maintain progress over
time. Integration of ABA principles into daily routines and activities is
crucial to promote generalization of skills beyond the therapy sessions. ABA
can be implemented across various environments, including homes, schools,
clinics, and community settings, to support individuals in developing
essential skills, such as communication, social interaction, self-help, and
academic abilities.

###  II-C Verbal Behavior Milestones Assessment and Placement Program (VB-
MAPP)

The Verbal Behavior Milestones Assessment and Placement Program (VB-MAPP) is a
criterion-referenced assessment based on typical language development, thus
measuring how well an individual performs against an objective rather than
another student. In practice, the tool provides an extensive checklist for
tracking progress over time. The VB-MAPP guidelines [11] suggest 18
categories, each with 15 objectives that must be completed in a predetermined
order. These categories are used to monitor the progress of ABA activities.

In this study, we focused on three categories that the therapists identified
as high priority: Tact, Listener, and Visual-Perceptual Matching to Sample
(VP-MTS). Tact is the ability to label or describe objects, actions, or events
in the environment. It involves expressing what one observes using language.
Listener skills refer to receptive language abilities where an individual
responds appropriately to verbal stimuli or instructions from others. It
includes following directions and understanding spoken information. VP-MTS
tasks involve matching or identifying objects, pictures, or symbols based on
visual cues, often used for learning and discrimination tasks.

##  III Related Work

There is a vast literature on the applications of serious game for autism
spectrum disorders that has been recently surveyed in [3, 4]. In this section,
we overview the most relevant mobile applications available on the marketplace
that are related to ABA therapy.

AutiSpark [12] is an educational mobile app designed for children with autism
spectrum disorder that provides many engaging and interactive learning games
tailored to meet the child’s learning needs. It covers many topics, such as
spelling, math, memory, sorting, matching, and puzzles. Its content spans
various concepts such as image association, emotional comprehension, sound
recognition, and much more. AutiSpark is exceptionally engaging and features a
variety of vibrant colors and animated elements on the screen. This
characteristic led the therapists to suggest that the app was designed for
children who have already made some progress in their ability to discern
between essential and non-essential elements. Furthermore, the app lacks a
clear differentiation between the various focus areas within ABA therapy.
AutiSpark is designed as an app to entertain autistic children while ensuring
they learn and enhance specific skills. However, it is not suitable to adhere
to the ABA therapy guidelines.

Autism BASICS [13] is a mobile app developed by a team of psychologists,
speech, behavioral, and occupational therapists. It aims to engage children
with autism and other special needs, enabling parents to work closely with
their children. Autism BASICS provides daily activities assigned by the
child’s therapist or parent, a library of learning-focused activities crucial
for the development of children with autism, and a parent’s corner with
educational videos and content to facilitate collaboration between parents and
their children. Parents or therapists assign tasks from the library,
specifying the time the child should dedicate to these daily activities.
Therapists can define multiple levels within a particular category for the
child to focus on. Each day, the child completes these tasks presented in an
engaging and enjoyable game format. Autism BASICS offers thousands of
activities categorized by significance in early childhood development for
children with autism or developmental delays. These activities cover alphabet,
spelling, math, and more, focusing on behavioral, academic, sensory,
independent/self-help, communication, and social skills. According to the
therapists at Spazio Autismo, some of these activities may be integrated into
ABA therapy.

Autism ABC [14] encompasses a range of essential features tailored to enhance
learning and development in autistic children. It offers a selection of
learning activities that cover cognitive, linguistic, and motor skills. The
app is highly adaptable, making it suitable for children of various ages and
autism spectrum levels. It ensures inclusivity by providing content that
aligns with different developmental stages. The ABC App places a strong
emphasis on data collection and analysis. It meticulously tracks and records
each child’s results for every activity, presenting them through user-friendly
charts and indexes. These data-driven insights are valuable for assessing a
child’s progress. The app incorporates the concept of positive reinforcement,
making learning engaging and enjoyable for children. This approach motivates
children to participate in activities actively. However, therapists noted that
it did not strictly adhere to the ABA therapy guidelines.

The applications we evaluated with the therapists have unique strengths but
also areas for improvement with respect to Spazio Autismo’s needs. AutiSpark
impresses with its personalized learning approach and diverse content, yet it
may benefit from increased data sharing and classification of games under the
naming of ABA therapy. Autism BASICS excels in providing comprehensive content
and user-friendliness but could enhance its adaptability and engagement.
Finally, ABC App distinguishes itself with data-driven insights, though
diversifying content and strengthening data anonymity could further improve
its appeal. None of these apps faithfully align with ABA therapy, a major
requirement for therapeutic use at Spazio Autismo. Accordingly, we designed a
framework that could meet such strict therapeutic needs.

##  IV The Framework

Our framework was designed in cooperation with the therapists working at
Spazio Autismo to help them transition from the current paper-based to a
digitally supported therapy. The framework had to align with ABA guidelines
faithfully, fully support the flow of therapeutic sessions, help therapists
design personalized sessions, and preparing the learning material. The paper-
based protocol makes it very difficult to record data about during the
sessions. Accordingly, the framework had to record all available session data,
store them securely, proving access using a web-based front-end with,
possibly, some analytic support.

###  IV-A Therapy Sessions

Patients at Spazio Autismo have been diagnosed with classic autism. They all
have high support needs, which means they face significant challenges in
social communication and interactions. Many of them may be non-verbal or have
minimal speech, and they encounter substantial difficulties in performing
daily tasks, such as self-care and vocational activities. Sessions are one-to-
one (patient-therapist) meetings that last for approximately one hour. These
sessions involve ABA activities that are closely monitored using the VB-MAPP
categories (Section II).

Therapists interact with the patients with a deck of cards. Each card features
a white background with a cartoonish image of an everyday object. The
therapist presents these cards to the patient, selecting them based on the
exercise being conducted and, when possible, the child’s interests (e.g.,
cards with images of vehicles for children who show curiosity towards cars).

Each VB-MAPP category consists of 15 objectives of increasing difficulty. An
objective is completed when the specified number of correct responses, as
defined by the therapists, is attained. These values are determined for each
category and each objective and are not dependent on the specific patient.
During the session, the therapist only records the correct responses by
marking them with checkmarks on an unstructured tracking paper sheet.
Incorrect responses, including incorrect answers and unanswered questions, are
not recorded.

Once the session is completed, the therapist has a sheet on which the correct
responses obtained from the patient are written down. In preparation for the
next session, the therapist reviews the sheet to avoid proposing the same
responses the patient completed in the previous sessions. This tracking
process must be repeated until that objective is completed. This tracking
process can span up to several months.

Upon completing the objective, the therapist shares the correct responses with
the supervisor and the patient’s family. To do this, the therapist transcribes
all the correct answers during the period required to complete the level into
a form. The file is then printed and handed to the family.

###  IV-B Framework Design

The design of our framework focused on faithfully replicating the paper-based
protocol already in place to maintain adherence to ABA principles and
facilitate the transition to a fully digitalized protocol. Leveraging the
technology’s possibilities, therapists envisioned replicating the game
experience of the deck of cards in a newly developed gaming app, introducing a
gamified component. Therapists asked for tools that could support them in
preparing the digital card deck used in the digital-based sessions. They also
required the possibility of recording children’s responses (with other
performance and timing data) during the sessions and generating the reports
shared with the family automatically. From the perspective of privacy and
security, therapists must have access only to their patients’ data. The
children’s identities must be kept anonymous accordingly, no personal
identification data is stored on the backend.

###  IV-C The Architecture

Figure 1 shows the architecture that comprises three components: the mobile
app, a website, and the backend, enabling the communication between these two.
The creation of the mobile app and the website became necessary to replicate
ABA therapy in the digital format. The mobile app is exclusively intended for
gaming by patients during their therapy sessions, and it is deliberately
designed to be minimal and free of stimuli that could impact children’s
ability to maintain concentration in their therapy activities. Thus, the
development of separate access via the web was necessary to provide the
therapists with a space to manage data before meeting with the patients and
after the session ended.

At the beginning of each session, the therapist logs into the mobile app using
the patient’s credentials using a tablet. The therapist can access the patient
data and view the planned and achieved objectives stored in the backend. While
the patient plays ABA games on the tablet during a session, the app records
the patient’s performance data in the backend. Once the session is completed,
the therapist can access the session data from the website. When the patient
completes an objective, the therapist can download the report from the website
to share with the supervisor and the family. The therapist can add or modify
the patient’s objectives from the website if necessary. At the moment of the
evaluation, the framework has ABA games for three VB-MAPP categories (Section
II-C) that the therapists identified as high priority: Tact, Listener, and
Visual-Perceptual Matching to Sample (VP-MTS).

Figure 1: Data flow diagram

##  V Experimental Results

We evaluated our framework over four months with 18 children (7 females and 11
males) 4-7 years old following the ABA therapy at the Spazio Autismo in
Mantova, Italy. Note that our evaluations included all the children who were
eligible to participate; other children were excluded due to the specific
requirements of their diagnoses. All the personal and therapeutic information
about the children involved in the evaluation remained undisclosed. For
research purposes, the therapists confirmed that (i) all the children involved
in the evaluation could be collectively categorized under the diagnosis of
classic autism; they all have high support needs, limited or no verbal
communication abilities, and experience significant obstacles in their day-to-
day activities. The patients are listed in the analysis with a unique
numerical identifier. Our evaluation focused on three main aspects: Patient
Focus During Therapy, Speed of Objectives Completion, and the Analysis of
Error Rates.

###  V-A Focus During Therapy

When using the paper-based protocol, children easily get distracted due to
several stimuli in their learning environment. Our application cannot remove
distractions entirely, however, therapists told us that children appeared more
focused and engaged. We can estimate attention and involvement during sessions
by quantifying the amount of time children spend actively engaged, computed as
the difference between the last and first answer. Figure 2 shows the average
time previously spent using the paper-based protocol (column Before) and our
framework (column After) in each category considered. Note that, we do not
report averages for the paper-based protocol since the therapists were not
collecting timing for each session; the reported data are based on their own
evaluation and session objectives. For the Tact category, our framework does
not significantly increase the time spent actively engaged, and the role of
digital support appears to be relatively neutral in increasing the focus
during therapy. In the other categories, when utilizing the tablet as a
supportive tool, children seem to become notably more engrossed and immersed
in the therapeutic activities. We discussed the results with the therapists,
who noted that Tact is the category where the children do not actively operate
on the tablet. Therefore, we recorded a slight difference between the paper-
based and the digital supports. On the other hand, categories (Listener and
VP-MTS) where children are more actively involved with the tablet during the
sessions show a 32% to 68% improvement in the amount of time patients spend
actively engaged.

Figure 2: Average time spent per session in each category; bars report the
standard error of the mean.

###  V-B Objectives Completion

The time needed to complete therapeutic objectives (goals) set by the
therapists is a critical indicator of improvement for ABA therapy. The
children involved in this evaluation have different starting levels in
different categories, as they may vary considerably depending on each patient
diagnosis. Accordingly, our evaluation focused on the number of additional
objectives the children could achieve in the given time frame.

For each patient, we collected, the set of completed Tact, Listener, and VP-
MTS objectives. We also analyzed the (anonymized) paper reports collected for
the same patients in the four months prior to the evaluation using the paper-
based protocol; for consistency, we analyzed the same VB-MAPP. In the previous
four months, 92 objectives were completed with an average of 5.11 objectives
per children. At the end of the evaluation, 119 therapeutic objectives were
accomplished using our framework with an average 6.11 objectives. Figure 4
shows the average percentage of objectives completed for each category while
Figure 4 reports the number of objectives completed by each patient for each
category. Note that the evaluation involved a limited number of patients at
different phases of their therapy, of different categories, over a short time
window. Accordingly, these results should be viewed solely as an assessment of
a positive trend, also perceived by the therapists during the sessions.

Figure 3: Percentage of completed objectives per category.

Figure 4: Number of objectives completed for each category by each patient.

###  V-C Analysis of Error Rates

Patients’ errors were not recorded during paper-based sessions as they would
distract the therapists from the patient. Our framework automatically records
all the patient’s interactions with the tablet, thus allowing therapists to
analyze the number of errors and the error rate each patient makes during a
session. We compute a patient’s error rate as,

$\psi=\frac{\\#ofErrors}{\\#ofRightAnswersRequired}$

If $\psi\leq 1$ then the patient made an equal or fewer number of errors
compared to the correct answers; if $\psi>1$, then the patient committed more
errors than the required number of correct answers. It is worth noting that
the objective is considered completed once the patient attains the prescribed
quantity of correct answers, regardless of the number of errors made in the
process. Figure 6 reports the average error rate for each category across all
patients. The three categories show substantially different error rates.
Figure 6 shows the average error rate in each category with the corresponding
range of variation. Tact exhibits the highest average error rate (2.05),
meaning that, for an objective requiring 50 correct answers, a patient would
be presented with approximately $\approx 150$ questions (50 answered correctly
and 100 answered incorrectly). Listener has an average error rate of 1.78, and
VP-MTS has 1.64. According to the therapists, the significantly higher average
error for Tact could be attributed to various factors. Tact primarily focuses
on enhancing the patient’s ability to identify and name objects, actions, and
events. This category often serves as one of the initial areas of intervention
for patients who are new to ABA therapy and may encounter substantial
challenges. Additionally, Tact games typically lack the captivating elements
or actions that can engage children, making it more challenging for them to
maintain their focus during these activities. In conclusion, Tact has
exhibited the least favorable results from the perspective of patient outcomes
among the three categories developed, coherently with the therapists’
expectations, which had never been quantified before.

Figure 5: Error rate of each patient in each category.

Figure 6: Error rate $\psi$ per category: mean with standard error (blue
line), minimum (green line), maximum (red line).

###  V-D Time Spent on Non-Therapeutic Activities

Autism Spectrum Disorder (ASD) patients are prone to being distracted. Before
the introduction of our app, many children lost their focus while therapists
were setting up the learning and data collection materials. Our framework
automates these two processes, and session setup is almost eliminated or
reduced to the bare minimum. According to the therapists, our application has
not only reduced their preparation and recording time but can also be
considered an element of improvement in therapy efficacy.

##  VI Conclusions and Future Developments

We presented a gamified framework we developed to assist therapists in
applying Applied Behavioral Analysis (ABA) therapy for autistic children. The
framework includes an application for tablets that therapists use during
therapeutic sessions with the children. The children’s therapeutic activities
are mini-games, providing safe and intuitive means to learn and improve in
various categories. The application also offers therapists an efficient way to
record several types of data automatically during the sessions. All the
recorded data and the sessions’ learning material can be accessed and prepared
using a web app that provides access to the recorded data and some editing
tools. We evaluated the framework at Spazio Autismo, with 18 children
diagnosed with classic autism, ranging from 4 to 7 years old. Our preliminary
results show that time spent on non-therapeutic activities has been reduced,
the patients’ focus increased by 10% to 68%, and speed of objectives
completion accelerated, with an average of 6.61 objectives completed in 4
months, in contrast with the 5.11 before the digitalization.

ABA therapy is heavily data-driven and the paper-based approach has posed
significant challenges. Our framework helps therapists collect more data that
can become integral to their analyses of patients’ progress. For example,
therapists could tailor the therapy to each patient, something infeasible with
paper-based therapy. Parents could use our framework outside the sessions to
offer children short, engaging games anytime during the day. This possibility
fits patients undergoing ABA therapy, which relies on the repetition of
behaviors.

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