Disentangling a Complex Response in Cell Reprogramming and Probing the Waddington Landscape by Automatic Construction of Petri Nets

Abstract

We analyzed the developmental switch to sporulation of a multinucleate Physarum polycephalum plasmodial cell, a complex response to phytochrome photoreceptor activation. Automatic construction of Petri nets from trajectories of differential gene expression in single cells revealed alternative, genotype-dependent interconnected developmental routes and identified metastable states, commitment points, and subsequent irreversible steps together with molecular signatures associated with cell fate decision and differentiation. Formation of transition-invariants in mutants that are locked in a proliferative state is remarkable considering the view that oncogenic alterations may cause the formation of cancer attractors. We conclude that the Petri net approach is useful to probe the Waddington landscape of cellular reprogramming, to disentangle developmental routes for the reconstruction of the gene regulatory network, and to understand how genetic alterations or physiological conditions reshape the landscape eventually creating new basins of attraction. Unraveling the complexity of pathogenesis, disease progression, and drug response or the analysis of attractor landscapes in other complex systems of uncertain structure might be additional fields of application.

True time series of gene expression from multinucleate single cells reveal essential information on the regulatory dynamics of cell differentiation

“…Petri nets, as they are used in this paper, contain one additional place C0, which does not represent a gene expression state. For simulation, C0 defines the initial gene expression state of the cell by randomly delivering its token to one of the places that are connected to C0 through socalled immediate transitions (filled in black) that fire immediately when the simulation starts (for details see (Rätzel et al 2020)). Connection to C0 also graphically highlights the places representing those gene expression states in which cell trajectories started.…”

“…The temporal sequences of gene expression patterns classified as Simprof significant clusters defined a trajectory for each individual cell and revealed significant differences between cell trajectories (Table 2). To relate gene expression states and trajectories we constructed a Petri net (bipartite graph) as previously described (Rätzel et al 2020;Werthmann and Marwan 2017), by representing each gene expression state by a place and the temporal transit between two states by a transition (Fig 4). A single token marking one place of the Petri net indicates the current gene expression state of a cell.…”

“…As each transition is connected to exactly two places (one preplace and one post-place), tokens are neither formed nor destroyed when moving through the net, so the gene expression state of the cell remains unequivocally defined at any time. The coherent Petri net obtained this way represents a state machine predicting possible developmental trajectories in terms of Markov chains of gene expression states (Rätzel et al 2020). The basic modelling principles are summarized in Table 3.…”

“…We have argued that Petri net modelling disentangles the complex response of cell reprogramming (Rätzel et al 2020) Table 2. Places and transitions in the two nets were colored according to different criteria.…”

“…Because cell cycle, cell fate choice, and development are synchronous throughout the plasmodium (Hoffmann et al 2012;Rätzel and Marwan 2015;Rusch et al 1966;Starostzik and Marwan 1995a;Walter et al 2013), single-cell gene expression trajectories can indeed be constructed from time series. By assembling finite state machines made from trajectories we have constructed Petri net models for the state transitions that predict Markov chains as variable developmental routes to differentiation (Rätzel et al 2020;Werthmann and Marwan 2017) which may be considered as trajectories through the Waddington landscape (Huang et al 2009;Waddington 1957). These Petri nets also predict reversible and irreversible steps, commitment points, and meta-stable states in cells responding to a differentiation stimulus.…”