Computational modeling of the immune response in multiple sclerosis using epimod framework

Background Multiple Sclerosis (MS) represents nowadays in Europe the leading cause of non-traumatic disabilities in young adults, with more than 700,000 EU cases. Although huge strides have been made over the years, MS etiology remains partially unknown. Furthermore, the presence of various endogenous and exogenous factors can greatly influence the immune response of different individuals, making it difficult to study and understand the disease. This becomes more evident in a personalized-fashion when medical doctors have to choose the best therapy for patient well-being. In this optics, the use of stochastic models, capable of taking into consideration all the fluctuations due to unknown factors and individual variability, is highly advisable. Results We propose a new model to study the immune response in relapsing remitting MS (RRMS), the most common form of MS that is characterized by alternate episodes of symptom exacerbation (relapses) with periods of disease stability (remission). In this new model, both the peripheral lymph node/blood vessel and the central nervous system are explicitly represented. The model was created and analysed using Epimod, our recently developed general framework for modeling complex biological systems. Then the effectiveness of our model was shown by modeling the complex immunological mechanisms characterizing RRMS during its course and under the DAC administration. Conclusions Simulation results have proven the ability of the model to reproduce in silico the immune T cell balance characterizing RRMS course and the DAC effects. Furthermore, they confirmed the importance of a timely intervention on the disease course.

Duplication. Considering the Teff duplication event we have to distinguish two possible cases: 1) the Teff symmetric duplication with probability ρ dup = 2/3 and a Teff asymmetric duplication, implying the T Memory effector differentiation, with probability ρ mem = 1 − ρ dup . This is modeled exploiting three different transitions: TeffDup Sym out( in) and TeffDup Asym out. In the CNS we model only the symmetric differentiation. The asymmetric differentiation give rise to the production of Effector memory cells. These last remain in blood circulation and are able to respond faster to antigen stimulation.The asymmetric differentiation occurs in peripheral blood circulation, then the produced effector memory cells can migrate to all other tissues (e.g. CNS). Where p T ef f Dup is the constant Teff duplication rate, see Table S1, andx(ν) = x T ef f in when the transition considered is the TeffDup Sym in, otherwisex(ν) =

So let define
x T ef f out .
Activation. The TeffActivation out and TeffActivation in transitions model the activation of the Teff cells in the peripheral lymphonode/blood vessel ( out) and the Central Nervous System (CNS) ( in). The activation of the Teff cells is modeled by the following functions where p T ef f Activation is the constant Teff activation rate, see Table S1, and x(ν) = {x RestingT ef f out , x Antigen , x IF N g out } when the transition considered is the . Let us note that the term (0.5 + exp(−x IF N g in( out) (ν)/Cif n)) is a coefficient varying in [0.5, 1.5] respectively to the concentration of INF γ , more is present into the system slower is the velocity. In particular when there is no INF γ then to the velocity is associated the highest value, (i.e., 1.5). otherwise it decreases until 0.5 .
Similarly, the transition MemActivation modeling the activation of T Memory effectors only in the peripheral lymphonode/blood vessel is defined as where and t 2inj is the time corresponding to the second antigen injection. We are considering the velocity of this transition as zero ∀ν < t 2inj , since we are assuming that the T Memory effectors start to react after the first virus occurrence, with twice the velocity of the Teff cells (for this reason we have 2 * p T ef f Activation ).
NK entry. The NKentry transition keeps in a constant range around 30 ( [3,4]) the number of N K out. It is defined by the following function: Killing. The general transitions modeling the killing of specific cell are the following: • the TregKillsTeff out and TregKillsTeff in modeling the controlling action of the Treg over the Teff cells. These are defined as follows: representing the Treg cell need fo te IL-10 for suppression of the Teff cells. Indeed, with an increasing number of IL-10 (i.e., the coefficient goes to 1) the transition velocity increases, otherwise it decreases. • The TeffkillsA and TeffKillsODC modeling the annihilation of the pathogen by the Teff action and the ODC damage due to Activated Teff cells, respectively, are defined as follows: with l ∈ cd(ODC), Let define Θ in(/out) as a coefficient varying in [0.5, 1.5] which takes in account the pro-(IL-17, IFNg) and anti-inflammatory (IL-10) cytokines in order to increase the velocity of the Teff action when more proinflammatory cytokines are present into the system, or decrease it otherwise. These coefficients are defined as follows: • Finally the Daclizumab action to control the Treg and Teff cells spreading is modeled by the transitions DACkillTeff and DACkillTreg, whose functions are defined as follows: where the coefficient 1 scales the velocity with respect to the number of Teff and Treg cells.
The parameters p Transition name are defined in Table S1.

S1.3 Data information
Referring to the Table 2 in the main paper, we related those numbers to 1 ul and identified the values to be included in our model. In particular, we infer the numbers of T effector cells from the sum of IFNg values and IL-17 values, and the the number of regulatory T cells from IL-10 values. Differently, to attribute a value for circulating cells in the CNS in healthy subjects we identified a threshold value from the literature and then attributed random values below this threshold [7].