Fig 1.
Flow-migration coupling within the A branch model.
An idealised model of a vessel bifurcation with shear stress differences present. (A) The network consists of a feeding vessel connected to a draining vessel by a short proximal path and a longer distal path. Blue arrows indicate the direction of flow throughout the network. Flow at the bifurcation near the inlet diverges, while flow near the outlet converges. The difference in path lengths results in different levels of flow/shear stress within each branch. (B) The network was discretised and seeded with an initial number of ECs (agents). Pressure boundary conditions (black) and values of flow (blue) and shear stress (red) are given in the initial configuration of the network. Periodic boundary conditions were prescribed at the inlet and outlet in order to keep the total number of cells within the simulation constant. In this model we are concerned with EC behaviour at flow-convergent bifurcations where two options to migrate against the flow exist: which path to the migrating ECs choose, and what determines this choice? (C) Vessel lumens are approximated in 3D by wrapping the number of cells in the vessel, n, each with width w, into the circumference of a circle. Flow and shear stress are then calculated using the Hagen-Poiseuille equation.
Fig 2.
Simple bifurcation rules (BRs 1 through 4).
The mean diameter was calculated over all segments composing the proximal branch (blue) and the distal branch (red). Snapshots of the network at various time points are presented on the far right of every panel. (A) Simulations using BR 1 always resulted in the regression of the distal branch and bifurcation loss, as all cells chose to enter the high-flow proximal branch. (B) In BR 2, cells always chose the path that requires the smallest change in migration direction, resulting in the loss of the proximal branch and the bifurcation. This simulation experiences numerous oscillations in diameter as the distal path is twice the length of the proximal path, and it takes several trips around before the smoothing algorithm settles down the diameter fluctuations (these oscillations have a period of roughly 30 time steps while the length of the distal path round-trip is 30 segments and agents move 1 segment at a time). (C) Mean diameter and standard deviation for the 10 runs using BR3. Using a random number generator to determine which branch each cell entered with equal probability resulted in stabilisation of both branches and no discernible difference in diameter between the two branches. (D) Mean diameter and results of each run using BR 4. Using unequal probability between the two branches, while favouring the high-flow branch, resulted in a form of diameter control, as the high-flow proximal branch stabilised at a larger diameter than the low-flow distal branch.
Fig 3.
Stability analysis of the weight parameter α in the mechanistic bifurcation rule.
The weight parameter α is used to scale the respective influence of the shear stress ratio and cell number ratio when calculating probability of a cell to enter each branch. We discretised α over its range. (A) On the left, the percentage of total regression events in all simulations and the percentage of those regression events that involved either the proximal or distal branch. On the right, the percentage of regression events over time. Setting α = 0.0 means only cell number was used to determine the probability of each branch; 91.9% of these simulations experienced regression and bifurcation loss, with 52% involving proximal regression vs. 48% distal (B) Setting α = 1.0 and using only shear stress to determine branch probability resulted in 100% of simulations losing the bifurcation, with 27% involving proximal regression vs. 73% distal. Additionally, these simulations were prone to lose the bifurcation much earlier, with near 100% regression reached after just 2 days of migration. (C) Contour plot of bifurcation loss over the whole range of α, which demonstrates a global minimum of stability for values of α ranging from 0.3 to 0.6. (D) The surface of bifurcation loss vs. α over time resembles and asymmetric saddle with similar rates of increase in bifurcation loss in both directions. (E) Bifurcation loss grouped within each day of migration. The majority of bifurcations were lost during day 2, while minimal loss occurred after that. (F) Mean diameter of the proximal (blue) and distal branch (red) with standard deviation for simulations within the stable region (α = 0.45). The high-flow proximal branch, on average, stabilised at a larger diameter than the low-flow distal branch. Note that both the proximal and distal branch experience some initial transients in cell number/diameter which eventually stabilise once cells have traversed each path completely. As a result, transients take longer to settle in the distal branch due to the longer path length.
Fig 4.
Competitive oscillations between the flow-based mechanism (shear stress) and collective-based mechanism (junction forces) achieves bifurcation stability.
(A) The probability of each branch of pulling in new cells at the bifurcation, Pi, is given by the weighted average of the shear stress probability Pτi and the cell number probability Pni. (B) The branch probability over time for an example stable simulation with a value of α = 0.45. The branch probability at the bifurcation oscillated slightly over time, with the probability of the high-flow proximal branch averaging to a higher value (dark blue, mean 0.6456) than the low-flow distal branch (dark red, mean 0.3544). (C) The individual components of these probabilities also oscillate over time in both the proximal branch (left) and the distal branch (right), with alternating peaks in shear stress (dark blue/red, solid line) and cell number probability (light blue/red, dashed-dot line). Additionally, a peak in either component in one branch was accompanied by a trough the that same probability component in the other branch. These data suggest that the shear stress and cell number components interact competitively at stable bifurcations, each compensating for increases in the other to prevent either of the two components from dominating the bifurcation (which would result in a loss of one of the two branches).
Fig 5.
The competition between shear stress and junction force transmission results from the inverse relationship between cell number and flow resistance.
The shear stress ratio within each vessel can also be expressed as a multiplicative combination of the cell number and pressure drop over the vessel. If we monitor cell number (red) and the pressure drop (black) in both the high-flow (top) and low-flow (bottom) vessel, we find that these two quantities are inversely correlated. Drops in the cell number are accompanied by spikes in the pressure drop and shear stress, as the narrowing of the vessel increases its resistance to flow. Similarly, increases in cell number result in a drop in resistance and the pressure drop, which decreases the amount of shear stress present in the vessel.
Table 1.
Glossary of constants and variables with the model, values and units.