Fig 1.
High-throughput time-lapse imaging of single cells migrating on 1D micropatterned lanes.
(a) Migration dynamics of single cells on one-dimensional Fibronectin (FN) lanes are recorded by scanning time-lapse measurements. FN lanes are fluorescently labeled, depicted here in green. At each point in time a bright field (BF) image showing the cell contour, and a DAPI fluorescence image indicating the position of the nuclei are captured. The FN lanes are automatically detected, the nuclei tracked, and the cells’ contours segmented. (b) Single cell trajectories, shown as position of cell front, back and nucleus plotted against time, reveal a broad spectrum of cell-type specific features. The first row depicts typical trajectories for MDA-MB-231 cells, the second row for MCF-10A cells. Horizontal scale bar represents 1h, vertical scale bar 100μm.
Fig 2.
Biophysical model of cell migration in one dimension.
(a) Cartoon of the mechanical protrusion competition model. The cell is defined by three marks: back, nucleus and front. Together the marks set the protrusion length L. Front and back are coupled to the nucleus by an effective elastic spring E and coupled to the ground by a non-linear molecular clutch κ. Polymerization forces F push the membrane outwards. The cell experiences the drag ζ during its movement. Redrawn from [13]. (b) Cartoon of the molecular clutch. Actin polymerization at the edge of the cell creates a retrograde flow vr. Talin-integrin mediated coupling (kon and koff) between the actin network and the fibronectin substrate B results in an effective friction force. Friction slows down the retrograde flow vr and pushes the membrane outwards.
Fig 3.
Simulation-based inference (SBI) of model parameters.
A schematic representation of the SBI workflow. A neural network is trained with simulated data, and subsequently experimental data are analyzed using the pretrained network. (1) A set of parameters {θ} is randomly sampled from the prior p(θ) and used to simulate trajectories x. x is a tensor with the shape (721, 3), where 721 represents the number of time points and 3 corresponds to spatial 1D coordinates: front, back, and nucleus. (2) The trajectory is downsampled into a low-dimensional feature space by a convolutional neural network (CNN). (3) The downsampled trajectory is fed into the neural density estimator (NDE) which outputs the posterior density. The log-likelihood of the NDE at the true point (θ) is used as a loss function to update the NDE. The trained NDE has a maximum likelihood at the true parameter point as marked by an ‘X’. The trained SBI is then used to estimate parameters from measured data. (5) The experimental trajectories are downsampled into a low-dimensional feature space by the same CNN as in step (2) and then fed into the previously trained NDE. (6) The resulting posterior represents a cell-specific parameter estimation describing interpretable properties of the cell as defined by the biophysical model.
Table 1.
Bounds of the uniform prior p(θ).
The 10 parameters and two noise amplitudes entering our biophysical model. 5 parameters are variable and the target of our inference procedure; one of the noise amplitudes is latent; all other parameters are kept constant. The bounds of the prior were chosen in agreement with the results of our previous publication Amiri et al. The prior was designed such that the parameter values of Amiri et al. fall within the prior. For the two noise amplitudes we chose values that produced results that were in quantitative agreement with experiments.
Fig 4.
Validation of SBI applied to a simulated trajectory.
A simulated trajectory x from parameter set θ and its corresponding posterior distribution. The posterior probability p(θ|x) was inferred by the trained NDE. On the right hand side the posterior distributions for each pair of parameters p(θij|x) are plotted (true parameters shown as white cross). The posterior distribution p(θi|x) for each individual parameter estimation is given by the blue graph on the diagonal with the true parameter value indicated by the vertical orange line. Horizontal gray lines in the plots on the diagonal represent a uniform posterior. The boundaries of the x-axis correspond to those of the prior p(θi) (see Table 1).
Fig 5.
Validation of SBI on experimental data.
Four different trajectories and posterior probabilities p(θ|x) as inferred by our trained NDE from experimental (left side) and simulated data (right side). The trajectories can be seen in the lower right corner. The posterior distribution for each individual parameter p(θi|x) are plotted on the diagonals and in the upper right corner the posterior distribution for each pair of parameters p(θij|x) are shown. The axes’ limits correspond to the prior limits as listed in Table 1. Left side in (a) and (b): typical 24h trajectories of MDA-MB-231 cells and their estimated posterior distribution. Vertical scale bars represent 100μm. Right side: the posterior distributions corresponding to each of the two experimental trajectories were used to sample parameters {θ}. The most likely parameter value from the experimental posterior distribution θtrue was chosen to simulate a trajectory x, and the simulated trajectory x used to estimate the posterior distribution again to verify our approach. The vertical orange lines on the histograms and the crosses in the density plots show θtrue.
Fig 6.
Comparative characterization of migratory phenotype for two cell lines.
(a, b) 10 randomly chosen trajectories for MDA-MB-231 and MCF-10 cells, respectively. (c) Ensemble posterior distribution of estimated model parameters using SBI. For each trajectory in a given population 1000 different points were sampled in parameter space. The plots show the ensemble average of all sampled points for all trajectories of a given population (NMDA = 85, NMCF = 30). Cell length L0 and actin polymerization rate Ve0 are the most distinctive parameters.
Fig 7.
Effect of inhibitors on model parameters as inferred by SBI.
Ensemble posterior distribution of model parameters of experimental cell trajectories using SBI. (a) Latrunculin A significantly decreases the actin network extension rate Ve0 for both MDA-MB-231 and MCF-10A cells while leaving all other parameters unchanged. (b) A similar effect can be observed for Y27632 treatment. Additionally, the resting protrusion length L0 is shifted towards larger values. For each trajectory in a given population we sampled 1000 different points in parameter space. Here, the ensemble of all sampled points for all trajectories of a given population is shown. MDA experiments, 5 replications, NMDA_ctrl = 85, NMDA_LatA = 129, NMDA_Y27 = 96; MCF experiments, 4 replications, NMCF_ctrl = 301, NMCF_LatA = 465, NMCF_Y27 = 507.