On logical bifurcation diagrams

Multistability and predominant hybrid phenotypes in a four node mutually repressive network of Th1/Th2/Th17/Treg differentiation

Elucidating the emergent dynamics of cellular differentiation networks is crucial to understanding cell-fate decisions. Toggle switch - a network of mutually repressive lineage-specific transcription factors A and B - enables two phenotypes from a common progenitor: (high A, low B) and (low A, high B). However, the dynamics of networks enabling differentiation of more than two phenotypes from a progenitor cell has not been well-studied. Here, we investigate the dynamics of a four-node network A, B, C, and D inhibiting each other, forming a toggle tetrahedron. Our simulations show that this network is multistable and predominantly allows for the co-existence of six hybrid phenotypes where two of the nodes are expressed relatively high as compared to the remaining two, for instance (high A, high B, low C, low D). Finally, we apply our results to understand naïve CD4+ T cell differentiation into Th1, Th2, Th17 and Treg subsets, suggesting Th1/Th2/Th17/Treg decision-making to be a two-step process.

Lattice structures that parameterize regulatory network dynamics

We consider two types of models of regulatory network dynamics: Boolean maps and systems of switching ordinary differential equations. Our goal is to construct all models in each category that are compatible with the directed signed graph that describe the network interactions. This leads to consideration of lattice of monotone Boolean functions (MBF), poset of non-degenerate MBFs, and a lattice of chains in these sets. We describe explicit inductive construction of these posets where the induction is on the number of inputs in MBF. Our results allow enumeration of potential dynamic behavior of the network for both model types, subject to practical limitation imposed by the size of the lattice of MBFs described by the Dedekind number.

Exploring attractor bifurcations in Boolean networks

Background

Boolean networks (BNs) provide an effective modelling formalism for various complex biochemical phenomena. Their long term behaviour is represented by attractors–subsets of the state space towards which the BN eventually converges. These are then typically linked to different biological phenotypes. Depending on various logical parameters, the structure and quality of attractors can undergo a significant change, known as a bifurcation. We present a methodology for analysing bifurcations in asynchronous parametrised Boolean networks.

Results

In this paper, we propose a computational framework employing advanced symbolic graph algorithms that enable the analysis of large networks with hundreds of Boolean variables. To visualise the results of this analysis, we developed a novel interactive presentation technique based on decision trees, allowing us to quickly uncover parameters crucial to the changes in the attractor landscape. As a whole, the methodology is implemented in our tool AEON. We evaluate the method’s applicability on a complex human cell signalling network describing the activity of type-1 interferons and related molecules interacting with SARS-COV-2 virion. In particular, the analysis focuses on explaining the potential suppressive role of the recently proposed drug molecule GRL0617 on replication of the virus.

Conclusions

The proposed method creates a working analogy to the concept of bifurcation analysis widely used in kinetic modelling to reveal the impact of parameters on the system’s stability. The important feature of our tool is its unique capability to work fast with large-scale networks with a relatively large extent of unknown information. The results obtained in the case study are in agreement with the recent biological findings.

Computational Verification of Large Logical Models-Application to the Prediction of T Cell Response to Checkpoint Inhibitors

At the crossroad between biology and mathematical modeling, computational systems biology can contribute to a mechanistic understanding of high-level biological phenomenon. But as knowledge accumulates, the size and complexity of mathematical models increase, calling for the development of efficient dynamical analysis methods. Here, we propose the use of two approaches for the development and analysis of complex cellular network models. A first approach, called "model verification" and inspired by unitary testing in software development, enables the formalization and automated verification of validation criteria for whole models or selected sub-parts. When combined with efficient analysis methods, this approach is suitable for continuous testing, thereby greatly facilitating model development. A second approach, called "value propagation," enables efficient analytical computation of the impact of specific environmental or genetic conditions on the dynamical behavior of some models. We apply these two approaches to the delineation and the analysis of a comprehensive model for T cell activation, taking into account CTLA4 and PD-1 checkpoint inhibitory pathways. While model verification greatly eases the delineation of logical rules complying with a set of dynamical specifications, propagation provides interesting insights into the different potential of CTLA4 and PD-1 immunotherapies. Both methods are implemented and made available in the all-inclusive CoLoMoTo Docker image, while the different steps of the model analysis are fully reported in two companion interactive jupyter notebooks, thereby ensuring the reproduction of our results.

Multi-parameter exploration of dynamics of regulatory networks

Over the last twenty years advances in systems biology have changed our views on microbial communities and promise to revolutionize treatment of human diseases. In almost all scientific breakthroughs since time of Newton, mathematical modeling has played a prominent role. Regulatory networks emerged as preferred descriptors of how abundances of molecular species depend on each other. However, the central question on how cellular phenotypes emerge from dynamics of these network remains elusive. The principal reason is that differential equation models in the field of biology (while so successful in areas of physics and physical chemistry), do not arise from first principles, and these models suffer from lack of proper parameterization. In response to these challenges, discrete time models based on Boolean networks have been developed. In this review, we discuss an emerging modeling paradigm that combines ideas from differential equations and Boolean models, and has been developed independently within dynamical systems and computer science communities. The result is an approach that can associate a range of potential dynamical behaviors to a network, arrange the descriptors of the dynamics in a searchable database, and allows for multi-parameter exploration of the dynamics akin to bifurcation theory. Since this approach is computationally accessible for moderately sized networks, it allows, perhaps for the first time, to rationally compare different network topologies based on their dynamics.