A Mechanistic, Stochastic Model Helps Understand Multiple Sclerosis Course and Pathogenesis

Hindawi Publishing Corporation
International Journal of Genomics
Volume 2013, Article ID 910321, 10 pages
http://dx.doi.org/10.1155/2013/910321
Research Article
A Mechanistic, Stochastic Model Helps Understand Multiple
Sclerosis Course and Pathogenesis
Isabella Bordi,1
Renato Umeton,2
Vito A. G. Ricigliano,2
Viviana Annibali,2
Rosella Mechelli,2
Giovanni Ristori,2
Francesca Grassi,3
Marco Salvetti,2
and Alfonso Sutera1
1
Department of Physics, Sapienza University of Rome, Piazzale Aldo Moro 2, 00185 Rome, Italy
2
Centre for Experimental Neurological Therapies (CENTERS), Neurology and Department of Neurosciences,
Mental Health and Sensory Organs, S. Andrea Hospital, Sapienza University of Rome, via di Grottarossa 1035-1039,
00189 Rome, Italy
3
Department of Physiology and Pharmacology, Sapienza University of Rome, Piazzale Aldo Moro 5, 00185 Rome, Italy
Correspondence should be addressed to Marco Salvetti; marco.salvetti@uniroma1.it and
Alfonso Sutera; alfonso.sutera@roma1.infn.it
Received 24 October 2012; Revised 2 January 2013; Accepted 27 January 2013
Academic Editor: Brian Wigdahl
Copyright © 2013 Isabella Bordi et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Heritable and nonheritable factors play a role in multiple sclerosis, but their effect size appears too small, explaining relatively little
about disease etiology. Assuming that the factors that trigger the onset of the disease are, to some extent, also those that generate
its remissions and relapses, we attempted to model the erratic behaviour of the disease course as observed on a dataset containing
the time series of relapses and remissions of 70 patients free of disease-modifying therapies. We show that relapses and remissions
follow exponential decaying distributions, excluding periodic recurrences and confirming that relapses manifest randomly in time.
It is found that a mechanistic model with a random forcing describes in a satisfactory manner the occurrence of relapses and
remissions, and the differences in the length of time spent in each one of the two states. This model may describe how interactions
between “soft” etiologic factors occasionally reach the disease threshold thanks to comparably small external random perturbations.
The model offers a new context to rethink key problems such as “missing heritability” and “hidden environmental structure” in the
etiology of complex traits.
1. Introduction
Multiple sclerosis (MS) is an immune-mediated disease of
the central nervous system with a relapsing-remitting course
in the majority of the early stages of the disease [1]. As
for other multifactorial diseases, there is no comprehen-
sive overview of the events that lead to the disease. This
limits the opportunities provided by the advancements in
genetics, immunology, and neurobiology since it is difficult
to contextualize each single discovery. The uncertainties
in the interpretation of genome-wide association studies
(GWAS) reflect, to some extent, this problem. These studies
carried the expectation to define the heritable component in
multifactorial diseases and, through this, also sketch the non-
heritable (environmental) contribution to the phenotype. As
largely witnessed by the debate about “missing heritability” in
multifactorial diseases, also this powerful approach appears
to be in need of interpretative keys as neither genes nor the
environment seem to harbour factors that, alone or jointly,
are strong enough to explain the disease etiology [2, 3].
Likewise situations are rather common in the physics
of nonlinear systems; here the observed large variations
are explained through the effects induced by small ran-
dom perturbations [4–6]. An example is the theory of the
Earth’s climate: the cooperative effect of a small stochastic
perturbation and periodic forcing (variation of astronomical
parameters) provides an amplification of the climate response
known as “stochastic resonance” that leads to the transition
from a temperate climate to an ice-covered Earth state and
vice versa [5–7].

2 International Journal of Genomics
Random molecular events can contribute to phenotypic
diversity in isogenic populations. Such variation can arise
from random fluctuations in gene expression [8, 9], and in
fact the term “gene expression noise” is typically used in
broad reference to variations among seemingly identical cells
experiencing the same environment [10]. Noise in genetic
circuits can have profound implications also in multicellular
organisms enabling physiological regulation mechanisms,
differentiation strategies, and facilitating adaptation and evo-
lution [11], often with a qualitatively different outcome than
a deterministic one [12]. As such, these discrete and random
fluctuations in the expression of individual genes may also
participate in the transition between health and disease by
exposing hidden genetic and environmental risk.
We show that, in MS, transitions between remissions
and relapses (i.e., states of relative health and disease, resp.)
indeed occur randomly suggesting that stochastic events
may contribute to disease pathogenesis. This is described
in a model that accommodates the erratic course of the
disease and concedes that “soft” heritable and nonheritable
perturbations can be amplified over time. The model is
based on the equation for a double-well forced by a random
perturbation (noise) [13]. We assume that the patient has two
states, disease (relapse) and health (remission), and that the
barrier to overcome for going from one state to the other
is determined by his genetic background and environmental
exposures. The small random perturbation is thought to be
the inherent variability in gene expression that triggers the
erratic sequence of relapses.
In the likely hypothesis that the mechanisms leading to
the transition between remission and relapse recapitulate
those that are responsible for the transition between health
and disease at onset, time and noise appear as crucial
variables that can “amplify” gene-environment interactions
and effects until, at one time or the other, the transition
towards the disease state occurs in “any” susceptible patient,
provided that enough observation time is given.
2. Materials and Methods
2.1. Patients Data. We used a dataset of relapses and remis-
sions of 70 patients (28 males and 42 females) with definite
MS [14, 15] who were monitored prospectively along the
years at the MS Clinic of Sapienza University of Rome (Italy)
by four experienced MS neurologists. All patients were free
of any disease-modifying therapy (all data are antecedent
1993) and had received short courses of corticosteroids if
deemed necessary during relapses. Times to disability end-
points were not different between this cohort and published
data on the natural history of the disease. The period of
interest starts with the first relapse at onset and ends with the
last one before the shift to a secondary progressive form. A
relapse was defined [14] as the occurrence of new symptoms,
the reappearance of former ones, or the worsening of current
symptoms of at least 24 hours duration but less than 6 month.
Its duration was calculated as the interval between onset of
the first symptom or sign and maximum improvement of the
last symptom or sign. As part of the standard procedures
of specialized MS Clinics, which require to firmly establish
the presence or absence of disease activity and progres-
sion in order to ensure the most appropriate therapeutic
interventions, symptoms were routinely verified during visits
at the outpatient service of the MS Clinic. Being this a
analysis of anonymous clinical data (collected until 1993,
before the institution of ethics committee in Italy in 1998),
stored for both clinical and research purposes in the database
of the University hospital, ethics committee approval and
written informed consent of patients are not needed
(http://www.garanteprivacy.it/garante/doc.jsp?ID=1884019).
Each subject has been represented by a sequence of plus
one (+1) and minus one (–1) corresponding to the states of
relapses and remissions, respectively. For clarity, hereafter
these two states are denoted as no health and health. Events are
reported on a weekly scale (shorter exacerbations have been
rounded up to one week).
2.2. The Model. First, the dataset is analyzed to study the
time evolution of relapses, their time duration, and their
statistical distribution. Secondly, a simple model consisting of
a mechanistic component and a stochastic perturbation [13]
is introduced to represent the main characteristics observed
in the clinical histories of patients. The mechanistic com-
ponent provides the deterministic (predictable) component
of the model, while the stochastic perturbation describes
an “external stimulus” whose behaviour is intrinsically non-
deterministic (random) and that can be analyzed only in
probabilistic terms. The latter is usually called “stochastic
forcing” since it acts as a force that perturbs the deterministic
component, making the model solution unpredictable.
The underlying hypothesis of the model here proposed
is that the health state of a patient in time behaves likewise
the motion of a particle of unit mass in a double-well energy
potential as schematically illustrated in Figure 1. Now, given
any initial position of the particle and if no perturbation
is applied on it (i.e., under deterministic conditions), the
particle will fall in one of the two potential wells and will
remain there for infinitely long time, independent of whether
the well is deep or flat (Figure 1(a)). If, instead, there is
a strong stochastic perturbation acting on the particle, it
will randomly move forth and back between the two wells,
independently of their depth. The most interesting situation
for us occurs when there is a small/moderate stochastic
perturbation acting on the particle: the particle will typically
stay for some time in the well it occupies, until the random
diffusion drives it over the potential barrier into the other well
(Figure 1(b)). It is intuitively clear that the exit from a flat well
happens faster than the exit from a deep well.
In the phenomena of diffusion, like the one just described,
the random perturbation is represented by the Wiener pro-
cess [16], which consists of a normally distributed white noise
(i.e., the amplitude of the noise is normally distributed with
given mean and standard deviation, and all frequencies are
involved with no time periodicity).
We note that the noise in physical systems was consid-
ered for a long time just as a nuisance that degrades the
signal. Later on it was found that, in complex (nonlinear)

International Journal of Genomics 3
Shallow well
Potential barriers
separating the wells
Deep well
(a)
Deep well
Shallow well
(b)
Figure 1: Schematic picture of a particle in a double-well energy potential. (a) The unperturbed particle falls into one of the two wells from its
initial position (deterministic case). (b) The particle subjected to a random perturbation jumps from a well to the other randomly (stochastic
case).
processes, as the two-state process here considered, it may
introduce a longitudinal dependence corresponding to the
erratic switching between the steady states that, otherwise,
would be persistent forever [5]. This property of noise
should be distinguished from the mechanism of “stochastic
resonance” introduced to explain the cyclic recurrence of ice
ages and that is based on the combined effect of noise and a
periodic external forcing [6]. In that case, useful applications
have been found in physical, technological, and biomedical
contexts ([17–20] and references therein).
Coming back to the metaphor particle-patient, the noise
may represent small biological variations of random charac-
ter experienced by the patient at each time. The bottom of the
two wells represents the health and no health states, while the
height of the wells is the barrier to be overcome for going from
one state to the other, here supposed to be determined by
the genetic background and environmental exposures of the
patient. Only the effect of noise over time can let the patient
change his state of health into disease and be subjected to
random relapses.
3. Results
3.1. Some Evidence from Clinical Data. By analyzing the
dataset of 70 patients, some evidence emerges. Figure 2 shows
the time evolution of the health states for three sample
patients. The time series suggest the unpredictable nature
of relapses (+1 states) followed by periods of relative quiet
remission with no new signs of disease activity (–1 states).
Also, the time in a health state seems much longer than
the one spent in a disease state. It is worth noting that the
transition between the two states, by construction, is here
represented by a step function (abrupt change of state). Such
a choice can be reductive or unrealistic since patients usually
report a slow or subacute onset of the relapses. However, this
feature does not affect the results presented below.
As shown in Figure 3, the duration of the relapsing-
remitting phase (i.e., period of analysis) varies from one
patient to the other, from a minimum of 40 weeks to a maxi-
mum of 1311 weeks (about 27 years). As illustrated by Figure 3,
for 25 of 70 patients (about 36%) the relapsing-remitting
phase lasts for about 200 weeks (about 4 years), and moving
towards longer periods the percentage of interested patients
quickly decreases to zero (exponential decaying distribution).
Similarly, the distributions of the time duration (or exit
time) of relapses and remissions follow exponential decaying
behaviours (Figures 4(a) and 4(b)). The mean duration of
disease attacks is about 4.3 weeks, with a minimum of 1
week to a maximum of about 24 weeks (rare events). The
mean time spent by patients in the health state is instead
about 100 weeks, with a minimum time interval between
two consecutive disease events of a few weeks up to about
1000 weeks in exceptional cases. It is worth noting that the
absence of any peak in the distributions at some specific time
corroborates what has been noted before for sample patients
(Figure 2): disease events occur randomly in time. Moreover,
the mean duration of disease events is much shorter than that
of remission.
In the following, we will show how these features can
be taken into account in a simple mechanistic model forced
by random perturbations. The aim is to contribute to the
understanding of the mechanisms that underlie the disease,
highlighting the role of noise in a system (patient) that is
primarily governed by mechanistic laws.
3.2. The Model: Mechanistic Component. The model to simu-
late the random transitions between the health and no health
states (–1 and +1, resp., as in clinical data) is here introduced.

0 50 100 150 200 250 300 350 400 450 500
1
Time (week)
Health
state
Nohealth
sate
−1
Patient no. 23
(a)
0 50 100 150 200 250 300 350 400 450 500
Time (week)
1
Health
state
Nohealth
sate
−1
Patient no. 32
(b)
0 50 100 150 200 250 300 350 400 450 500
1
Health
state
Nohealth
sate
−1
Time (week)
Patient no. 53
(c)
Figure 2: Time evolution of health and no health states for three sample patients taken from the set of 70 patients. The health state of each
patient has been represented by a sequence of plus one (+1) and minus one (−1) corresponding to the states of relapses and remissions,
respectively. Events are reported on a weekly scale and the shift to a secondary progressive form of the disease has been excluded.
40 199 358 517 676 835 994 1153 1312
0
5
10
15
20
25
30
Period of analysis (week)
Counts(numberofpatients)
Figure 3: Histogram of the duration of the relapsing-remitting
phase in weeks for the 70 selected patients. The duration of the
relapsing-remitting phase (i.e., analysis period) varies from one
patient to the other, from a minimum of 40 weeks to a maximum of
1311 weeks, following an exponential decaying distribution. For 25 of
70 selected patients (about 36%), the relapsing-remitting phase lasts
for about 200 weeks (about 4 years).
Let 𝑥 be the health state of a patient (the position of the
particle in our metaphor). For a two-state process, as the one
we are interested in, the time evolution of 𝑥 is given by the
following equation:
𝑑𝑥 = 𝑥 (1 − 𝛼𝑥2
) 𝑑𝑡, (1)
where 𝑡 is the time, 𝛼 is a control parameter, and 𝑑𝑥 and
𝑑𝑡 are the infinitesimal variation of the health state and
of time, respectively. The above equation has three steady
(independent on time) states. They are 𝑥0 = 0, 𝑥1 = −1/√𝑎,
and 𝑥2 = +1/√𝑎. Note that in case 𝛼 = 1, we have 𝑥1 = −1
and 𝑥2 = 1. Now, in agreement with the convention used for
clinical data, let 𝑥1 be the state of health and 𝑥2 the state of no
health.
The potential 𝑉, associated with (1) can be easily com-
puted [6] as,
𝑉 (𝑥) = − ∫ 𝑥 (1 − 𝛼𝑥2
) 𝑑𝑥 = −
1
2
𝑥2
+ 𝛼
1
4
𝑥4
. (2)
It can be noted that 𝑉(𝑥) is a symmetric function with respect
to a change of sign of the 𝑥 variable (it is the same for 𝑥 greater
than zero or 𝑥 less than zero), describing a double-well.
For clarity, we show in Figure 5 the potential 𝑉 as a
function of 𝑥 for two values of the control parameter: 𝛼 = 1
and 𝛼 = 0.7. It should be noted that the difference of the
potential in 𝑥1 (𝑥2) and 𝑥0 (denoted by Δ𝑉 in the figure) is
the barrier that must be overcome to jump from one state to
the other; it increases when 𝛼 is decreased with respect to the
reference value 𝛼 = 1 (the opposite occurs when 𝛼 is greater
than 1). In the context of MS, such a barrier is though to be
set by the combination of heritable and nonheritable risk.
Thus, according to (1) and (2), we have
(a) if, by any chance, at initial time the patient is nearby
𝑥1, he will be there forever;
(b) if, by any chance, at initial time the patient is nearby
𝑥2, he will be there forever;

0
20
40
60
80
100
120
140
1 3.5 6 8.5 11 13.5 16 18.5 21 23.5 25
Duration of no health events (week)
Counts(no.ofnohealthevents)
No health events: total = 218
Duration of no health events: mean = 4.3weeks
(a)
0 200 400 600 800 1000 1200
0
50
100
150
200
Duration of health events (week)
Counts(no.ofhealthevents)
Health events: total = 266
Duration of health events: mean = 100weeks
(b)
Figure 4: Histograms of the time duration of relapses and remissions for the 70 selected patients. (a) Histogram of no health events (+1 states);
(b) histogram of health events (−1 states). Both figures show exponential decaying distributions with no sign of periodicity (i.e., peak at given
duration), suggesting that disease events occur randomly in time. To be noted is the different duration of relapses and remissions (on average
4.3 weeks against about 100 weeks).
−1 0 1 2
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
𝑥
𝑉
𝛼 = 1
𝑥0
𝑥1 𝑥2
Δ𝑉
−2
(a)
−2 −1 0 1 2
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
𝑥
𝑉
𝛼 = 0.7
Δ𝑉
𝑥0
𝑥2𝑥1
(b)
Figure 5: Potential 𝑉 (double-well) as a function of the state variable 𝑥. The potential 𝑉 associated with the mechanistic model described
by (1) is plotted for two values of the parameter 𝛼 controlling the height of the wells: (a) 𝛼 = 1 and (b) 𝛼 = 0.7. The variable 𝑥 represents the
health state of a patient (i.e., the position of the particle in our metaphor). The health state is denoted by 𝑥1, the no health state by 𝑥2, and 𝑥0 is
the unstable steady state between them. Δ𝑉 is the barrier whose height is controlled by the parameter 𝛼. Dashed lines denote the upper and
lower bounds of the potential barrier.
(c) if, by any chance, at initial time the patient is nearby
𝑥0, he will have 50% chance to be forever nearby 𝑥1 or
𝑥2.
Now, as shown by clinical data (Figure 2), the patient (𝑥)
stays a longer time in the health state compared to the no
health state. This feature suggests that the depth of the two
wells differs and, in particular, the one associated to the health
state is deeper than the other, so that it is more difficult
to exit from the health condition than from the relapse. In
our mathematical model, this implies the introduction of an
asymmetry in 𝑉(𝑥), as
𝑉 (𝑥) = − ∫ [𝑥 (1 − 𝛼𝑥2
) − 𝛽] 𝑑𝑥 = −
1
2
𝑥2
+ 𝛼
1
4
𝑥4
+ 𝛽𝑥,
(3)
where the parameter 𝛽 determines the magnitude and shape
of the asymmetry.

Figure 6 illustrates an example of asymmetric double-
well (𝛽 = 0.08) for 𝛼 = 1 and 𝛼 = 0.7. In Figure 6,
the barriers are denoted as Δ𝑉1 and Δ𝑉2 and are the
difference in the potential between the states 𝑥1 and 𝑥0, and
𝑥2 and 𝑥0, respectively. In all the three circumstances above,
such barriers separate the states 𝑥1 and 𝑥2, preventing the
switching between the wells. Now the question is: can some
external random stimuli provide enough energy for diffusing
𝑥 across the barriers, regardless the initial conditions?
3.3. The External Random Stimuli. Unless new risk factors are
discovered, those identified so far in MS appear unsuitable as
deterministic triggers of continuous and random transitions
between health and disease, inducing very different disease
courses not only in different patients but also in the same
patient at different time points. As noise is a source of
variability in biological systems, we tried to model it as a
driving force. This kind of forcing must be specified through
its statistical properties. Let us suppose that on average the
noise has zero effect (i.e., its time mean is zero), has a given
variance, and it is time decorrelated (i.e., at a given time the
value of the stimulus depends only on the previous time): this
is the zero order approximation of the statistics of noise (i.e.,
the Wiener process previously discussed). By introducing
such external random perturbation into (1), we have
𝑑𝑥
change of the health
state of a patient
= [𝑥 (1 − 𝛼𝑥2
) − 𝛽] 𝑑𝑡
mechanistic component
(double-well)
+ 𝜀1/2
𝑑𝑤,
external random
stimuli
(4)
where 𝜀1/2
𝑑𝑤 is the noise of variance 𝜀. Because of the exter-
nal random stimuli, this equation has important deviations
from its deterministic version described by (1): now the
patient can change randomly his state from health to no health
and vice versa.
Averaged over time it is impossible that 𝑥 crosses the
barrier since the noise, by construction, has zero mean.
However, given the statistical property of the stimulus, in due
time, we expect that a sequence of small driving occurrences
may be equally signed so that 𝑥 may overcome the barrier
and switch from one state to the other. In other words,
𝑥 is harvesting energy from the stimuli for crossing the
barrier. For the sake of clarity, the relationship among disease
characteristics and model parameters are summarized in
Table 1.
As we shall see, this model fits nicely the statistics of the
exit times of disease events and explains a few paradoxes
encountered in understanding the disease course.
3.4. Model Solutions. To better understand the model, let
us look at the solutions of (4) for different values of the
parameters. Setting 𝛼 = 1, the solution of (4) as a function of
time is shown in Figure 7 for the symmetric and asymmetric
double-well cases (𝛽 = 0 and 𝛽 = 0.08, resp.). According to
our metaphor, the patient alternates periods of wellness and
disease randomly and, as expected, when a small asymmetry
is taken into account, the patient spends more time in the
health state than in the no health state. This means that there
is a finite probability that for any initial condition nearby 𝑥1
(or 𝑥2) the patient will jump across 𝑥0 at a finite time 𝜏 (i.e.,
time duration in a given state), switching from one state to
the other.
As seen from Figure 4, the duration time of each state, 𝜏,
is a random variable. The mean value of 𝜏 in the health state
(𝑥1) can be estimated [6] and is given by
𝜏𝑥1 ≈ 𝑒2Δ𝑉1/𝜀
(5)
while in the no health state (𝑥2) is
𝜏𝑥2 ≈ 𝑒2Δ𝑉2/𝜀
, (6)
where 𝑒 is the Nepero number. It is interesting to note that,
knowing the mean values of duration times in the health and
no health states, it is possible to estimate the ratio between
Δ𝑉1 and Δ𝑉2, say the asymmetry of the double-well.
For the dataset discussed in the previous section, we have
𝜏𝑥1 ≈ 100 weeks and 𝜏𝑥2 ≈ 4.3 weeks. On the other hand,
taking the logarithm of (5) and (6), and performing their
ratio, we get
Δ𝑉1
Δ𝑉2
≈
log (𝜏𝑥1)
log (𝜏𝑥2)
. (7)
Thus, the ratio of the logarithms of duration times in the
states 𝑥1 and 𝑥2 provides an estimate of the ratio between the
barriers that must be overcome to move from one state to the
other. By considering the duration times 𝜏𝑥1 and 𝜏𝑥2 averaged
over all the cases of 70 patients it is found Δ𝑉1/Δ𝑉2 ≈ 3.1.
This implies that the well in 𝑥1 is about three times deeper
than that in 𝑥2.
We point out that the ratio between the depths of the wells
can be estimated for each patient, provided that the mean
duration times of the two states are known from the clinical
history. As an example, we have estimated such a ratio for the
three sample patients considered in Figure 2. We found
𝜏𝑥1 ≈ 117.7 weeks, 𝜏𝑥2 ≈ 1.5 weeks, Δ𝑉1/Δ𝑉2 ≈ 11.8,
for the patient no. 23;
for the patient no. 32;
for the patient no. 53.
At this stage, the potential 𝑉 associated with each patient can
be estimated by fixing 𝛼 = 1 and changing 𝛽 so that the
two wells are asymmetric in the predetermined ratio. Plots of
the three potentials are shown in Figure 8. By assuming that
random stimuli of the same variance are applied to the three
patients, the shapes of the potentials reflect the differences in
the duration times of health and disease states observed in
Figure 1. The first potential has a very deep well associated
with the health state and a very shallow well associated with
the no health state: this configuration lets the patient be
affected by a few disease attacks of short duration. Given the
same observation period, the patient no. 32 is expected to

−1 0 1 2
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
𝑥
𝑉
𝑥0
𝑥1
𝑥2
−2
𝛼 = 1, 𝛽 = 0.08
Δ𝑉1
Δ𝑉2
(a)
𝛼 = 0.7, 𝛽 = 0.08
Δ𝑉1
Δ𝑉2
𝑥0
𝑥1
𝑥2
−1 0 1 2
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
𝑥
𝑉
−2
−0.8
(b)
Figure 6: Asymmetric potential 𝑉 as a function of the state variable 𝑥. The asymmetry in the depth of the wells is obtained setting 𝛽 ̸= 0
in (3). Here the effect of 𝛽 (𝛽 = 0.08) on the shape of the potential 𝑉 is illustrated for the same values of the control parameter 𝛼 used in
Figure 5: (a) 𝛼 = 1 and (b) 𝛼 = 0.7. Δ𝑉1 and Δ𝑉2 are the barriers between 𝑥1 (𝑥2) and 𝑥0, respectively. Dashed lines denote the upper and lower
bounds of the potential barriers.
Table 1: Disease characteristics and parameters of the mechanistic stochastic model. The table summarizes analogies and links between MS
characteristics (left column) and model parameters in (4), in the right column.
Disease characteristics during the relapsing-remitting phase Mechanistic, stochastic model
Two states are observed that we referred to as health and no health.
𝑉(𝑥): double-well potential (two minima associated with the two
states).
The disease develops in patients with a certain degree of heritable
and nonheritable risk.
𝛼: control parameter that sets the barrier to overcome for going
from one state to the other.
It is found that the time spent in the state of health is much longer
than in the no health state.
𝛽: asymmetry parameter of the potential.
Random variability of biological processes (i.e. randomness in
transcription and translation leading to cell-to-cell variations) [8].
𝜀1/2
𝑑𝑤: stochastic perturbation with zero mean and variance 𝜀.
Genetic background and environmental factors alone are not
enough to trigger the transitions between the two states.
The noise provides the required energy for the random transitions.
have more attacks compared to the previous patient due to
the difference in the depth of the wells. Lastly, the patient no.
53 when compared to the patient no. 32 should experience
disease events lasting more time because of the deeper well
associated with the no health state.
4. Discussion
The contribution of the heritable or non-heritable factors
that drive MS onset and course appear uniformly too small
to explain its etiology, variable severity, and erratic mani-
festations. We present a mechanistic, stochastic model that
describes how the effects of the above forces may be randomly
amplified over time. Given a time of observation T (e.g.,
the average lifespan of a person), patients may experience a
transition between the two states with a mean time 𝜏. Thus,
the probability to observe a transition requires that T is much
longer than 𝜏. However, since 𝜏 is randomly distributed, by
mere chance, the transition may or may not occur within the
observation time T. Within this approach, time is conceived
not only as a random variable characterizing disease events
but also as the “observation time” needed to detect the
onset of the disease in “any susceptible” patient. Stochastic
noise in gene expression, through its pervasive effects on
virtually all biological processes, may be the factor that
amplifies and reshapes the deterministic effects of genetic and
environmental risk factors.

0 100 200 300 400 500 600 700 800 900 1000
0
1
2
−2
−1
𝑥
𝛼 = 1, 𝛽 = 0, 𝜀 = 0.13
Time
(a)
0 100 200 300 400 500 600 700 800 900 1000
0
1
2
Time
−2
−1
𝑥
𝛼 = 1, 𝛽 = 0.08, 𝜀 = 0.13
(b)
Figure 7: Solutions of the mechanistic, stochastic model. The solutions of (4) as a function of time are shown for (a) the symmetric (𝛽 = 0)
and (b) asymmetric (𝛽 = 0.08) double-well potential. In both cases, the control parameter is set to 𝛼 = 1, and a stochastic perturbation of the
same variance is applied (𝜀 = 0.13). Under the effect of the random perturbation, the state variable 𝑥 (i.e., the health state of a patient in our
metaphor) jumps erratically from −1 to +1 and vice versa (a). When a small asymmetry is introduced in the potential 𝑉, the variable 𝑥 still
changes erratically its state but spends more time around −1 than around +1 (b). The latter case appears suitable for describing the random
occurrences of relapses in MS as observed in clinical data.
−1 0 1 2
−0.8
−0.6
−0.4
−0.2
0.2
0.4
0.6
0.8
𝑥
𝑉
−2
0
Patient no. 23
𝛼 = 1, 𝛽 = 0.25
(a)
−1 0 1 2
−0.8
−0.6
−0.4
−0.2
0.2
0.4
0.6
0.8
𝑥
𝑉
−2
0
Patient no. 32
𝛼 = 1, 𝛽 = 0.19
(b)
−1 0 1 2
−0.8
−0.6
−0.4
−0.2
0.2
0.4
0.6
0.8
𝑥
𝑉
−2
0
Patient no. 53
𝛼 = 1, 𝛽 = 0.12
(c)
Figure 8: Potential 𝑉 for three sample patients. As an example, the three patients in Figure 2 are considered (no. 23, no. 32 and no. 53 in our
dataset) and the mean duration of their relapses and remissions estimated. These values are then used to compute the ratio of the potential
barriers Δ𝑉1/Δ𝑉2 according to (7) (for the estimates see Model Solutions). The potential 𝑉 for each patient ((a)–(c)) is given by (3) setting 𝛼
= 1 and choosing 𝛽 so that the two wells are asymmetric in the predetermined ratio. In particular, it is found (a) 𝛽 = 0.25, (b) 𝛽 = 0.19, and (c)
𝛽 = 0.12. Dashed line denotes the upper bound of the potential barriers.
The model is in accord with a variety of data in MS
and may be general enough to explain features of other
complex traits. First, time is a key variable in this model
but also in experimental autoimmunity where there is evi-
dence of a multistep process made of subtle alterations—
resulting from quantitative trait loci variations—that may
accumulate with time and end up in susceptibility when
a threshold for the occurrence of the disease is passed
[21, 22]. Furthermore, defects of the immune response that
originate from predisposing genetic variants are present
before any onset of autoimmunity [23]. These relatively
recent observations are in accord with classical epidemio-
logical data suggesting that fairly long time intervals are
a necessary factor for the development of MS [24, 25].
Lastly, time itself remains an essential component of the
diagnostic algorithm of MS [26]. It is therefore not sur-
prising that, if time is a crucial factor to understand the
nature of the disease in diagnostic terms, it may also be
instrumental for the pathogenetic understanding of the
process.

Second, our model links the degree of genetic variation
with the probability that the transition between health and
disease states occurs over time. In fact, MS risk alleles
influence age of onset [27] and correlation of age of onset
in MS relative pairs is proportional to genetic sharing [28].
Interestingly, the model applies also when individuals share
genetic protection as witnessed by our observation of an
underrepresentation of MS among Italian twins, possibly due
to protective factors shared by twins in a Mediterranean
area [29]. Furthermore, as expected, a link between genetic
variation and transition between the two states over time
seems to be maintained during the disease course: with one
exception [30], the vast majority of the studies on the natural
history of MS has shown that, while the occurrence of relapses
is unpredictable, their frequency and severity tend to be
conserved as a longer first interattack interval is associated
with a better prognosis [31–34].
While this is the first time that a model combining
deterministic factors and stochastic forcing is applied to
the interpretation of a multifactorial disease, determinism
and stochasticity have been already shown to fit basic bio-
logical mechanisms ([8–12] and references therein). Here,
stochasticity is beneficial to “the individual, the colony,
or the species” [35] in that it may exert a physiological
regulatory role. It may therefore be feasible to seek strategies
that try to revert a “pathogenic stochasticity” to its original
physiologic role. In the stochastic resonance, introduced to
explain Earth’s glaciations [5] and sometimes erroneously
associated with the random transitions illustrated above,
the model solution is driven by the cooperative effect of
the stochastic perturbation and the small periodic forcing,
leading to a flip from one state to the other with the
frequency proper of the external periodic forcing; the small
stochastic perturbation provides the energy necessary for
the transition, while the periodic forcing sets the periodicity
of the transitions. The stochastic resonance occurs for low
frequency of the periodic forcing, say for long time periods.
For climate changes that induce glaciation cycles [6], such
a period is set to 100000 years that describes the long-term
variation of the energy input prescribed by astronomical
theory. In the case of relapsing-remitting MS, as clinical data
suggest, there is not a typical periodicity of relapses induced
by some external factors. However, we can imagine to exploit
a small external cyclic component in order to “stabilize” the
solution around a given state (the health state) “deamplifying”
it with some short periodicity, contrary to what happens
in the stochastic resonance, in order to prevent relapses.
A nonmutually exclusive approach may be to decrease the
variance of the noise 𝜀 (possibly acting on fluctuations of gene
expression).
This study offers a new interpretative key of the events
that characterize the clinical course of MS. By inference, it
also suggests that time and noise amplify and adjust the
deterministic influence of genes and environment on the
disease. It will not be easy to obtain direct experimental
evidence for this model at the disease level. However, basic
science data—increasingly supporting the notion that noise
provides critical functions at the cell level—suggest that this
effort will probably be a rewarding one. Thus far the model
may contribute to the understanding of phenotypic variation
in health and disease states.
5. Conclusion
Despite the advent of genomewide association studies
(GWAS; an approach to look for associations between hun-
dreds of thousands genetic variations and diseases), many
questions remain about the causative mechanisms of multi-
factorial diseases. In particular, the effect size of the genetic
variants identified through these studies explains relatively
little of the heritability (the proportion of total variance
in a population for a particular measurement, taken at a
particular time or age, that is attributable to variation in
genetic values) of most complex traits. So far, the same is true
for environmental risk factors. Multiple sclerosis is one of the
diseases that best exemplify this key problem. By observing
the course of this disease in its relapsing-remitting phase
we found that a mechanistic model with a random forcing
describes the main features of the disease course. If the model
is applied to the events that precede the disease (i.e., those
that derive from the effects of the genetic and environmental
risk factors), it may explain how relatively small effects may
be amplified by comparably small random perturbations. As
such, the model may inform the design of future studies on
gene-environment interactions in multifactorial diseases.
Authors’ Contribution
I. Bordi and R. Umeton contributed equally to the work and
should be considered co-first authors.
Acknowledgments
This work was supported by Fondazione Italiana Sclerosi
Multipla (FISM) and by Progetto Strategico 2007, Italian Min-
istry of Health. The funders had no role in study design, data
collection and analysis, decision to publish, or preparation of
the paper.
References
[1] A. Compston and A. Coles, “Multiple sclerosis,” The Lancet, vol.
372, no. 9648, pp. 1502–1517, 2008.
[2] T. A. Manolio, F. S. Collins, N. J. Cox et al., “Finding the missing
heritability of complex diseases,” Nature, vol. 461, no. 7265, pp.
747–753, 2009.
[3] E. E. Eichler, J. Flint, G. Gibson et al., “Missing heritability and
strategies for finding the underlying causes of complex disease,”
Nature Reviews Genetics, vol. 11, no. 6, pp. 446–450, 2010.
[4] G. E. P. Box and G. M. Jenkins, Time Series Analysis: Forecasting
and Control, Holden-Day, San Francisco, Calif, USA, 1970.
[5] A. Sutera, “On stochastic perturbation and long-term climate
behaviour,” Quarterly Journal, Royal Meteorological Society, vol.
107, no. 451, pp. 137–151, 1981.
[6] R. Benzi, G. Parisi, A. Sutera, and A. Vulpiani, “A theory
of stochastic resonance in climatic change,” SIAM Journal on
Applied Mathematics, vol. 43, no. 3, pp. 565–578, 1983.

[7] I. Bordi and A. Sutera, “Stochastic perturbation in meteorology,”
Waves in Random Media, vol. 10, no. 3, pp. R1–R30, 2000.
[8] A. Raj and A. van Oudenaarden, “Nature, nurture, or chance:
stochastic gene expression and its consequences,” Cell, vol. 135,
no. 2, pp. 216–226, 2008.
[9] A. Raj, S. A. Rifkin, E. Andersen, and A. van Oudenaarden,
“Variability in gene expression underlies incomplete pene-
trance,” Nature, vol. 463, no. 7283, pp. 913–918, 2010.
[10] D. Fraser and M. Kærn, “A chance at survival: gene expres-
sion noise and phenotypic diversification strategies,” Molecular
Microbiology, vol. 71, no. 6, pp. 1333–1340, 2009.
[11] A. Eldar and M. B. Elowitz, “Functional roles for noise in genetic
circuits,” Nature, vol. 467, no. 7312, pp. 167–173, 2010.
[12] N. Maheshri and E. K. O’Shea, “Living with noisy genes:
how cells function reliably with inherent variability in gene
expression,” Annual Review of Biophysics and Biomolecular
Structure, vol. 36, pp. 413–434, 2007.
[13] I. Bordi and A. Sutera, “Drought variability and its climatic
implications,” Global and Planetary Change, vol. 40, no. 1-2, pp.
115–127, 2004.
[14] G. A. Schumacher, G. Beebe, R. F. Kibler et al., “Problems of
experimental trials of therapy in multiple sclerosis: report by
the panel on the evaluation of experimental trials of therapy in
multiple sclerosis,” Annals of the New York Academy of Sciences,
vol. 122, pp. 552–568, 1965.
[15] C. M. Poser, D. W. Paty, L. Scheinberg et al., “New diagnostic
criteria for multiple sclerosis: guidelines for research protocols,”
Annals of Neurology, vol. 13, no. 3, pp. 227–231, 1983.
[16] N. Wiener, “Differential-space,” Journal of Mathematics and
Physics, vol. 2, pp. 131–174, 1923.
[17] J. K. Douglass, L. Wilkens, E. Pantazelou, and F. Moss, “Noise
enhancement of information transfer in crayfish mechanore-
ceptors by stochastic resonance,” Nature, vol. 365, no. 6444, pp.
337–340, 1993.
[18] R. Rouse, S. Han, and J. E. Lukens, “Flux amplification using
stochastic superconducting quantum interference devices,”
Applied Physics Letters, vol. 66, no. 1, pp. 108–110, 1995.
[19] F.-G. Zeng, Q.-J. Fu, and R. Morse, “Human hearing enhanced
by noise,” Brain Research, vol. 869, no. 1-2, pp. 251–255, 2000.
[20] F. Moss, L. M. Ward, and W. G. Sannita, “Stochastic resonance
and sensory information processing: a tutorial and review of
application,” Clinical Neurophysiology, vol. 115, no. 2, pp. 267–
281, 2004.
[21] C. C. Goodnow, “Multistep pathogenesis of autoimmune dis-
ease,” Cell, vol. 130, no. 1, pp. 25–35, 2007.
[22] J. H. Cho and P. K. Gregersen, “Genomics and the multifactorial
nature of human autoimmune disease,” The New England
Journal of Medicine, vol. 365, no. 17, pp. 1612–1623, 2011.
[23] L. Menard, D. Saadoun, I. Isnardi et al., “The PTPN22 allele
encoding an R620W variant interferes with the removal of
developing autoreactive B cells in humans,” The Journal of
Clinical Investigation, vol. 121, no. 9, pp. 3635–3644, 2011.
[24] J. F. Kurtzke, “Multiple sclerosis: changing times,” Neuroepi-
demiology, vol. 10, no. 1, pp. 1–8, 1991.
[25] A. E. Handel, G. Giovannoni, G. C. Ebers, and S. V. Ram-
agopalan, “Environmental factors and their timing in adult-
onset multiple sclerosis,” Nature Reviews Neurology, vol. 6, no.
3, pp. 156–166, 2010.
[26] C. H. Polman, S. C. Reingold, B. Banwell et al., “Diagnostic
criteria for multiple sclerosis: 2010 revisions to the McDonald
criteria,” Annals of Neurology, vol. 69, no. 2, pp. 292–302, 2011.
[27] S. Sawcer, G. Hellenthal, M. Prinen et al., “Genetic risk and a
primary role for cell-mediated immune mechanisms in multiple
sclerosis,” Nature, vol. 476, no. 7359, pp. 214–219, 2011.
[28] A. D. Sadovnick, I. M. Yee, C. Guimond, J. Reis, D. A. Dyment,
and G. C. Ebers, “Age of onset in concordant twins and other
relative pairs with multiple sclerosis,” American Journal of
Epidemiology, vol. 170, no. 3, pp. 289–296, 2009.
[29] G. Ristori, S. Cannoni, M. A. Stazi et al., “Multiple sclerosis in
twins from continental Italy and Sardinia: a nationwide study,”
Annals of Neurology, vol. 59, no. 1, pp. 27–34, 2006.
[30] M. Debouverie, S. Pittion-Vouyovitch, S. Louis, and F. Guillem-
in, “Natural history of multiple sclerosis in a population-based
cohort,” European Journal of Neurology, vol. 15, no. 9, pp. 916–
921, 2008.
[31] M. P. Amato, G. Ponziani, M. L. Bartolozzi, and G. Siracusa, “A
prospective study on the natural history of multiple sclerosis:
clues to the conduct and interpretation of clinical trials,” Journal
of the Neurological Sciences, vol. 168, no. 2, pp. 96–106, 1999.
[32] D. Miller, F. Barkhof, X. Montalban, A. Thompson, and M.
Filippi, “Clinically isolated syndromes suggestive of multiple
sclerosis—part I: natural history, pathogenesis, diagnosis, and
prognosis,” The Lancet Neurology, vol. 4, no. 5, pp. 281–288,
2005.
[33] A. Degenhardt, S. V. Ramagopalan, A. Scalfari, and G. C. Ebers,
“Clinical prognostic factors in multiple sclerosis: a natural
history review,” Nature Reviews Neurology, vol. 5, no. 12, pp.
672–682, 2009.
[34] A. Scalfari, A. Neuhaus, A. Degenhardt et al., “The natural
history of multiple sclerosis, a geographically based study 10:
relapses and long-term disability,” Brain, vol. 133, no. 7, pp. 1914–
1929, 2010.
[35] R. Losick and C. Desplan, “Stochasticity and cell fate,” Science,
vol. 320, no. 5872, pp. 65–68, 2008.

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