A Novel Hybrid Logic-ODE Modeling Approach to Overcome Knowledge Gaps

Mathematical modeling allows using different formalisms to describe, investigate, and understand biological processes. However, despite the advent of high-throughput experimental techniques, quantitative information is still a challenge when looking for data to calibrate model parameters. Furthermore, quantitative formalisms must cope with stiffness and tractability problems, more so if used to describe multicellular systems. On the other hand, qualitative models may lack the proper granularity to describe the underlying kinetic processes. We propose a hybrid modeling approach that integrates ordinary differential equations and logical formalism to describe distinct biological layers and their communication. We focused on a multicellular system as a case study by applying the hybrid formalism to the well-known Delta-Notch signaling pathway. We used a differential equation model to describe the intracellular pathways while the cell–cell interactions were defined by logic rules. The hybrid approach herein employed allows us to combine the pros of different modeling techniques by overcoming the lack of quantitative information with a qualitative description that discretizes activation and inhibition processes, thus avoiding complexity.


Model Parametrization Estimation of translation and transcription rates
To estimate maximal translation rate of Delta ( ) we used the equilibrium assumption. The external Delta input represents concentration of Delta on the adjacent cells, available for binding with Notch, and it can be either 0 or . In absence of Hes, Delta would reach the stable value . In this case, at equilibrium, maximal translation rate of Delta ( ) should satisfy the following condition:

= µ ⋅
Translation and maximal transcription rates of Hes, as well as the rate of NICD cleavage upon Delta binding, are found in the computational model of Agrawal et al., 2009.

Estimation of degradation rates
For each variable we assumed exponential decay, with constant degradation rates. They are computed from molecule half-life 1/2 , using: Ilagan et al. (Ilagan et al., 2011) estimates a half-life of 180 minutes for Notch intracellular domain thus Delta degradation rate is computed from its half-life of 4.9 hr (Preuße et al., 2015). Hes mRNA and protein half-lives are taken from Hirata et al., 2002 andKobayashi et al., 2015: 24.1 and22.3 min, respectively.

Transcription processes Hill equations
In any reaction where there is a binding-unbinding process, as per inducible/repressible promoters Hill equations properly depict the complex formation.

= +
Hill equations are characterized by two parameters: or dissociation constant and n the order. The latter usually identify the number of binding sites.
In our model we obtained from literature mining the maximum rate of transcription for Delta and Hes proteins, to represent their transcriptional regulation we multiplied this maximum rate by a Hill equation: Nucleus Import-Export: Passive and Active Transport Proteins of small dimensions, namely smaller than 60-70 kDa, can move through the nuclear envelope by passive diffusion. Big proteins, on the contrary, require nuclear localization signal (NLS) to be imported into the nucleus. NLS is an aminoacidic sequence which binds to nuclear transport receptors (importins), that mediates the transport of the protein into the nucleus. Nuclear export signal (NES) sequence, analogously, regulates export of a protein, by binding with exportins receptors.
Notch intracellular domain has a molecular weight of 110 kDa (Kelly et al., 2007). It is endowed of nuclear localization signal, while it does not carry NES. Following Cardarelli et al., 2008, which describes transport of a molecule of same size, we model the translocation of NICD in the nucleus, considering passive diffusion and active transport, mediated by NLS, only in entrance.  Figure   S1), expressing flow through the envelope in function of protein concentration were fitted, obtaining maximal velocity and Michaelis-Menten parameter . Since NICD has concentration at most of order 10 −3 µ ≪ = 73.24 µ , we can make the following approximation of the Michaelis-Menten term: Figure S1: Michaelis-Menten fit of the active import transport. Concentration of a specific protein can be computed as

Estimation of Protein Concentrations Using Proteomic Data
where φ is the size weighted abundance, expressed in ppm, and is the primary sequence length of the protein.

Wnt crosstalk
To help understanding the possible application of an independent input we integrated our Delta Notch model with a concentration dependent inhibition of the Notch activity. Due to the lack of experimental data, we used it as a scaling factor that multiplies the rate of Hes-mRNA production

Hybrid strategy pseudocode
Below we report the detailed pseudocode of the hybrid strategy simulation algorithm: Inputs: N ODE-modules with output variable V and independent input ( ), logical rule ( ), break condition, We select the module generating the event and select its time variable to synchronize all the module variables, by interpolation on the same integration steps (variable step algorithms generate different time arrays) ← ( ) = 1: We then delete all the integration performed after = 1:

( > )
We then restart the process end Per each iteration a single event is generated with a consequent update of the grid. The module variables are always synchronized to the first event in order to generate an output in which all the variables share the same time points for the integration.