Fig 1.
Illustration of model configurations and workflow of simulations.
(a) Cells are initialized as a monolayer with four endothelial cells (red) representing cross-section of blood vessels perpendicular to the plane of simulation. (b) Simulation workflow. In the initial regime nutrients are allowed to diffuse into the system to generate a steady-state distribution, during which cells metabolize without changing their size or phenotype. After the fields have stabilized, the simulation is started and is run in iterative cycles of Monte Carlo steps (MCS). After each MCS, each cell performs metabolic trafficking, and is probed for cell division. Upon division, a potential mutation event is triggered in the daughter cells. Inset: unbiased parameters from 104 cells after t = 1, 5, 10, …, 5 × 104 number of mutations are distributed uniformly in the algorithm.
Fig 2.
Simulations without mutations and with a stable nutrient source result in a stable tissue.
(a) Cell configuration at time t = 3000 MCS, showing pressure as color. Note the lack of any specific pattern in the distribution of intracellular pressure. (b) The number of cells, and intracellular pressure stay relatively constant in the course of the simulations. Cells cover approximately the whole of the tissue (coverage). Data averaged from 10 independent simulation runs. (c) Nutrient distribution in the system happens via diffusion from active blood vessels. The color codes the oxygen concentration in the system at t = 50,000 MCS. (d) Nutrient levels stay constant in the system with abundant glucose, somewhat less oxygen, and no lactate. The absence of lactate indicates that cells in the population remain respiratory.
Table 1.
List and values of mutating parameters in the model.
Fig 3.
Simulations with mutating cells and stable vasculature.
Mutation rate = 10%. (a-b) Simulations initiated with non-mutating cells (green) and a single mutating cell (red) near (a) or far from (b) the nutrient source. The frames are from simulations where the mutating cell persisted and took over the population. (c-f) Results from simulations initiated with mutating cells. (c) Cell configuration at time t = 3000 MCS, showing intracellular pressure as the color code. Note the fluctuations in pressure distribution compared to the non-mutating case in Fig 2. (d) Population dynamics of a mutating population as a function of simulation time, showing the number of cells, intracellular pressure and tissue coverage. The number of cells initially increases and after a peak drops to almost half of the peak value, where it stabilizes. The intracellular pressure follows the initial increase in cell numbers. During the cell number drop the pressure is moderated to almost half, after which it is increasing to reach a plateau. The tissue coverage drops from the initial full coverage to approximately 50%. (e) Oxygen concentrations in the system at t = 3000 MCS. Nutrients are depleted further from the blood vessels, eventually resulting in a shortage of oxygen in distant regions. (f) Nutrient levels during the simulation reveal the depletion of glucose and oxygen during the expansion phase. A simultaneous increase in lactate levels shows the appearance of fermentors. When nutrients are depleted, the population size declines. With the decrease of respiration, oxygen levels return to the level observed in the non-mutating populations (Fig 2d). Plots show average of 10 independent simulations.
Fig 4.
Stages of development in simulations with mutating cells and stable vasculature.
Mutation rate = 10%. (a) Number of cells and nutrient levels as in Fig 3f, using a logarithmic time-scale with the three stages indicated with numbers. (b) Cell configurations of each stage with oxygen concentrations color coded. (c) Population size (pink) and averages of all 10 evolving parameters scaled with their respective step sizes σp and shifted to their initial values. Intracellular growth signal: N0. (d) Normalized parameter values at the end of the simulation runs (t = 105 MCS). (e-f) Stage 1: expansion. Configuration of cells from a simulation showing the instantaneous growth rate (e) defined as the increase in target volume in the current MCS, and generation age (f) at t = 2200 MCS. Patches of high growth appearing independently from the localization of sources. (g-i) Stage 2: hypoxia. Configuration of cells from a simulation showing the intracellular growth signal N0 (g), level of hypoxia (h), and glucose uptake (i) at t = 6000 MCS. (j-k) Stage 3: starvation. Configurations of cells from a simulation showing growth rates (j) and intracellular pressure (k) at late stages of development. (l-m) Evolution of cellular metabolism showing intracellular hypoxia and ROS (l) and nutrient uptake (m). Line graphs show average of 10 independent simulations with standard deviation, box plots show median with interquartile range and minimum / maximum values from 10 independent simulations.
Fig 5.
Effect of mutation rate on population size and stage progression.
(a) Number of cells in the population as a function of time. (b-f) Nutrient levels in simulations with mutation rates 0.1% (b), 0.5% (c), 1% (d), 5% (e), and 10% (f) showing a similar behavior in all cases. (g) Distribution of cells along the three main principal components of the populations in phenotype space from example simulations with 1%, 10%, and 20% mutation rates at the end of simulations (t = 105 MCS). Each dot represents a single cell. (h) Relative weight (eigenvalues) of the principal axes of the populations shown in (g). (i) Composition of the first three principal axes in the populations shown in (g). (j-k) Spread and density of clusters identified in phenotype space at the end of simulations using hierarchical clustering, depicted as the function of distance from the origin. Each dot corresponds to one cluster from a total of 10 independent simulations. (l-m) Spread and density of clusters throughout the evolution of the model. Data from 10 independent simulation repeats from each condition, except (g-i) which shows data from one simulation from each condition.
Fig 6.
Effect of fluctuating nutrient supplies on the tissue.
(a) Number of cells at the end of simulations (t = 100,000 MCS), showing for simulations with different vessel tortuosities () and different cell mutation rates (μ). In simulations with healthy vessels (high switching probability
) a mutating cell population produces more cells than non-mutating ones. However, in simulations with erratic nutrient supply (with lower blocking probability
), the mutating populations die out more frequently than the non-mutating ones. At extreme erratic switching (
) 10 out of 10 mutating populations are extinct and even healthy ones start to die out (2 out of 10 simulations). Error bars indicate standard deviation of cell numbers from 10 repetitions. Values belonging to different simulations with the same mutation rate are connected and are slightly shifted on the x-axis for better visibility. (b) Average intracellular growth signal evolution in the population are not changed in simulations with different vessel blocking probabilities (mutation rate μ = 0.1%). This shows that the vessel blocking probability does not directly affect selection and progression speed. (c) The average intracellular growth signal in simulations with decreasing vessel blocking probability (
, solid lines) in mutating (red, μ = 10%) and not mutating (blue, μ = 0%) populations. Cell-cell competition in mutating populations initially drives the growth parameters to a high value just as in the simulations with stable nutrient sources. When the blocking probability reaches the magnitude of
and below, populations die out (dashed lines: percentage of simulations with living cells). (d) Population sizes in simulations with stable nutrient sources (
) and in simulations where the blocking probability is controlled by the cell density (
), creating a feedback. In both cases the populations survive. The tissue coverage is approximately halved, making the vessel blocking probability approximately
in the feedback simulations.
Fig 7.
(a) Clonal expansion. Model configurations of the same population at t = 1500 MCS and t = 6000 MCS. Each cell is depicted with one of 256 different colors at time t = 1500 MCS, and the daughters inherit this color from that time point onwards. The population 4500 MCS later consists of the descendants of about a dozen of the initial cells. (b) Cell trajectories showing that cells flow outwards from the nutrient sources (indicated with black dots). (c) Trajectories from a setup with randomly placed vessels, showing that the outflow is determined by the placement of vessels. (d) The population segregates into independently evolving parts around the blood vessels. Color code shows the chemotaxis parameter for glucose (χg(i, t)) and the stiffness parameter λv(i, t) at t = 70 000 MCS. Cells around the same source are homogeneous, but are different from cells surrounding other sources. (e) Chemotaxis parameter (χg(i, t)) and stiffness parameter (λv(i, t)) values in the populations shown in (d) as the function of time. The four segregated parts clearly separate in their parameter values, and evolve almost independently. Vertical line denotes the time t = 70 000 MCS corresponding to the time of the configurations on (d). (e) Trajectories from a late population in the fluctuating system, showing the trajectories moving from one source to another. Movement direction is indicated with arrows. (g, h): Combined chemotaxis parameters (〈χ′(i, t)〉i = 〈χg(i, t) + χo(i, t) − χl(i, t)〉i) as a function of time in simulations with different vessel tortuosities at the first 20 000 MCS (g) and for the whole simulation time (h). The parameter is increasing in all simulation conditions, showing that nutrient detection is important for cell survival. In simulations with lower blocking probabilities the random selection introduced by the blocking and opening of the vessels results in a higher noise in the combined chemotaxis parameters average. Values averaged from 10 independent simulation runs for each condition.