Dynamic modelling of signalling pathways when ordinary differential equations are not feasible

Abstract

Motivation: Mathematical modelling plays a crucial role in understanding inter- and intracellular signalling processes. Currently, ordinary differential equations (ODEs) are the predominant approach in systems biology for modelling such pathways. While ODE models offer mechanistic interpretability, they also suffer from limitations, including the need to consider all relevant compounds, resulting in large models difficult to handle numerically and requiring extensive data.

Results: In previous work, we introduced the retarded transient function (RTF) as an alternative method for modelling temporal responses of signalling pathways. Here, we extend the RTF approach to integrate concentration or dose-dependencies into the modelling of dynamics. With this advancement, RTF modelling now fully encompasses the application range of ODE models, which comprises predictions in both time and concentration domains. Moreover, characterizing dose-dependencies provides an intuitive way to investigate and characterize signalling differences between biological conditions or cell types based on their response to stimulating inputs. To demonstrate the applicability of our extended approach, we employ data from time- and dose-dependent inflammasome activation in bone marrow-derived macrophages treated with nigericin sodium salt. Our results show the effectiveness of the extended RTF approach as a generic framework for modelling dose-dependent kinetics in cellular signalling. The approach results in intuitively interpretable parameters that describe signal dynamics and enables predictive modelling of time- and dose-dependencies even if only individual cellular components are quantified.

Availability and implementation: The presented approach is available within the MATLAB-based Data2Dynamics modelling toolbox at https://github.com/Data2Dynamics and https://zenodo.org/records/14008247 and as R code at https://github.com/kreutz-lab/RTF.

Figure 1.

Illustrations of how dose dependence of dynamic parameters translates into kinetics. See Supplementary Table S3 for a complete overview. (a) Hill curve showing the dose dependency of amplitude A. (b) Resulting kinetics for the three values of A highlighted in a. For better clarity, only the sustained part is displayed and the response saturates to the value of A. (c) The dose-dependency of the response time τ follows a decreasing Hill curve. (d) The three highlighted values of τ in c impact the time shift τ of the RTF response curves.

Figure 2.

Comparison of fitting individual functions to each dose (single-dose RTF approach) with a joint fit of the dose-dependent joint model on the same experimental data. In contrast to the 35 parameters of the individual fits, the joint model requires only 11 parameters to describe the data and enables predictions for unobserved doses.

Figure 3.

The condition-specific dose-dependent RTF can describe the data generated over two different conditions (wild-type cells and NEK7 knockout cells). Furthermore, fold-changes Δ can be estimated and tested for significance.

Figure 4.

The profile likelihood of the fold-change parameters Δ can be used to test, which parameters are significantly different in the knockout condition. The intersection points between the dotted threshold and the profile likelihood indicate the margins of the 95% confidence interval (CI) highlighted by grey colouring. If the 0 on the logarithmic scale (corresponding to 1 on the linear scale) is outside the 95% CI, the likelihood ratio test is significant to a 5% significance level, indicating significantly different characteristics of the dose-dependency between both conditions.