Extracting cellular automaton rules from physical Langevin equation models for single and collective cell migration

Probabilistic cellular automata modelling of intercellular interactions in airways: complex pattern formation in patients with chronic obstructive pulmonary disease

Chronic obstructive pulmonary disease (COPD) is a highly prevalent lung disease characterized by chronic inflammation and tissue remodeling possibly induced by unusual interactions between fibrocytes and CD8+ T lymphocytes in the peribronchial area. To investigate this phenomenon, we developed a probabilistic cellular automata type model where the two types of cells follow simple local interaction rules taking into account cell death, proliferation, migration and infiltration. We conducted a rigorous mathematical analysis using multiscale experimental data obtained in control and disease conditions to estimate the model's parameters accurately. The simulation of the model is straightforward to implement, and two distinct patterns emerged that we can analyse quantitatively. In particular, we show that the change in fibrocyte density in the COPD condition is mainly the consequence of their infiltration into the lung during exacerbations, suggesting possible explanations for experimental observations in normal and COPD tissue. Our integrated approach that combines a probabilistic cellular automata model and experimental findings will provide further insights into COPD in future studies.

BIO-LGCA: A cellular automaton modelling class for analysing collective cell migration

Collective dynamics in multicellular systems such as biological organs and tissues plays a key role in biological development, regeneration, and pathological conditions. Collective tissue dynamics—understood as population behaviour arising from the interplay of the constituting discrete cells—can be studied with on- and off-lattice agent-based models. However, classical on-lattice agent-based models, also known as cellular automata, fail to replicate key aspects of collective migration, which is a central instance of collective behaviour in multicellular systems. To overcome drawbacks of classical on-lattice models, we introduce an on-lattice, agent-based modelling class for collective cell migration, which we call biological lattice-gas cellular automaton (BIO-LGCA). The BIO-LGCA is characterised by synchronous time updates, and the explicit consideration of individual cell velocities. While rules in classical cellular automata are typically chosen ad hoc, rules for cell-cell and cell-environment interactions in the BIO-LGCA can also be derived from experimental cell migration data or biophysical laws for individual cell migration. We introduce elementary BIO-LGCA models of fundamental cell interactions, which may be combined in a modular fashion to model complex multicellular phenomena. We exemplify the mathematical mean-field analysis of specific BIO-LGCA models, which allows to explain collective behaviour. The first example predicts the formation of clusters in adhesively interacting cells. The second example is based on a novel BIO-LGCA combining adhesive interactions and alignment. For this model, our analysis clarifies the nature of the recently discovered invasion plasticity of breast cancer cells in heterogeneous environments.

Pattern formation during embryonic development and pathological tissue dynamics, such as cancer invasion, emerge from individual intercellular interactions. In order to study the impact of single cell dynamics and cell-cell interactions on tissue behaviour, one needs to develop space-time-dependent on- or off-lattice agent-based models (ABMs), which consider the behaviour of individual cells. However, classical on-lattice agent-based models also known as cellular automata fail to replicate key aspects of collective migration, which is a central instance of collective behaviour in multicellular systems. Here, we present the rule- and lattice-based BIO-LGCA modelling class which allows for (i) rigorous derivation of rules from biophysical laws and/or experimental data, (ii) mathematical analysis of collective migration, and (iii) computationally efficient simulations.

Modelling collective cell motion: are on- and off-lattice models equivalent?

Biological processes, such as embryonic development, wound repair and cancer invasion, or bacterial swarming and fruiting body formation, involve collective motion of cells as a coordinated group. Collective cell motion of eukaryotic cells often includes interactions that result in polar alignment of cell velocities, while bacterial patterns typically show features of apolar velocity alignment. For analysing the population-scale effects of these different alignment mechanisms, various on- and off-lattice agent-based models have been introduced. However, discriminating model-specific artefacts from general features of collective cell motion is challenging. In this work, we focus on equivalence criteria at the population level to compare on- and off-lattice models. In particular, we define prototypic off- and on-lattice models of polar and apolar alignment, and show how to obtain an on-lattice from an off-lattice model of velocity alignment. By characterizing the behaviour and dynamical description of collective migration models at the macroscopic level, we suggest the type of phase transitions and possible patterns in the approximative macroscopic partial differential equation descriptions as informative equivalence criteria between on- and off-lattice models. This article is part of the theme issue 'Multi-scale analysis and modelling of collective migration in biological systems'.

NudCL2 regulates cell migration by stabilizing both myosin-9 and LIS1 with Hsp90

Cell migration plays pivotal roles in many biological processes; however, its underlying mechanism remains unclear. Here, we find that NudC-like protein 2 (NudCL2), a cochaperone of heat shock protein 90 (Hsp90), modulates cell migration by stabilizing both myosin-9 and lissencephaly protein 1 (LIS1). Either knockdown or knockout of NudCL2 significantly increases single-cell migration, but has no significant effect on collective cell migration. Immunoprecipitation–mass spectrometry and western blotting analyses reveal that NudCL2 binds to myosin-9 in mammalian cells. Depletion of NudCL2 not only decreases myosin-9 protein levels, but also results in actin disorganization. Ectopic expression of myosin-9 efficiently reverses defects in actin disorganization and single-cell migration in cells depleted of NudCL2. Interestingly, knockdown of myosin-9 increases both single and collective cell migration. Depletion of LIS1, a NudCL2 client protein, suppresses both single and collective cell migration, which exhibits the opposite effect compared with myosin-9 depletion. Co-depletion of myosin-9 and LIS1 promotes single-cell migration, resembling the phenotype caused by NudCL2 depletion. Furthermore, inhibition of Hsp90 ATPase activity also reduces the Hsp90-interacting protein myosin-9 stability and increases single-cell migration. Forced expression of Hsp90 efficiently reverses myosin-9 protein instability and the defects induced by NudCL2 depletion, but not vice versa. Taken together, these data suggest that NudCL2 plays an important role in the precise regulation of cell migration by stabilizing both myosin-9 and LIS1 via Hsp90 pathway.

Cellular automaton models for time-correlated random walks: derivation and analysis

Many diffusion processes in nature and society were found to be anomalous, in the sense of being fundamentally different from conventional Brownian motion. An important example is the migration of biological cells, which exhibits non-trivial temporal decay of velocity autocorrelation functions. This means that the corresponding dynamics is characterized by memory effects that slowly decay in time. Motivated by this we construct non-Markovian lattice-gas cellular automata models for moving agents with memory. For this purpose the reorientation probabilities are derived from velocity autocorrelation functions that are given a priori; in that respect our approach is “data-driven”. Particular examples we consider are velocity correlations that decay exponentially or as power laws, where the latter functions generate anomalous diffusion. The computational efficiency of cellular automata combined with our analytical results paves the way to explore the relevance of memory and anomalous diffusion for the dynamics of interacting cell populations, like confluent cell monolayers and cell clustering.