PMC

Snapshots of IBM GC B cell simulation. All B cells were started in the lower zone (DZ) and undergo spontaneous periodic motion between DZ and LZ as predicted by the phenomenological model. Chemokine concentrations in the DZ and LZ are volume-rendered, and the cell color shows density of CXCR4 receptors as indicated by the color bar (Color figure online)

Schematic showing model for regulation of chemokine receptor levels on the cell surface. Unbound receptors (U) and bound receptors (B) levels are determined by association or dissociation with chemokine (shown as red ellipse) with rates k ⁺ and k ⁻, respectively. Bound receptors are endocytosed to become internalized receptors (I) with rate μ and internalized receptors can either be degraded with rate δ or recycled with rate β. Finally, new receptor synthesis occurs with rate π (Color figure online)

Diversity of oscillations with the phenomenological model. In each case, initial conditions were x=0,v=1,r ₁=1,r ₂=1. Top row: Left panel shows symmetrical oscillations, center panel shows asymmetrical oscillations and right panel shows oscillations confined to one zone. Bottom row: Left panel shows “nested” large and small oscillations in both zones, center panel shows “nested” small oscillations only in one zone, and right panel shows chaotic oscillations. Parameters: Top left (c=10, w=25, π ₁=0.15, π ₂=0.15, τ ₁=0.06, τ ₂=0.06, δ ₁=0.006, δ ₂=0.006, κ ₁=0.5, κ ₂=0.5, ϵ ₁=0.3, ϵ ₂=0.3, χ=28, γ=5, k=50); top center (c=10, w=25, π ₁=0.23, π ₂=0.15, τ ₁=0.06, τ ₂=0.06, δ ₁=0.006, δ ₂=0.006, κ ₁=0.5, κ ₂=0.5, ϵ ₁=0.3, ϵ ₂=0.3, χ=28, γ=5, k=50); top right (c=10, w=25, π ₁=0.15, π ₂=0.15, τ ₁=0.06, τ ₂=0.06, δ ₁=0.006, δ ₂=0.006, κ ₁=3.6, κ ₂=0.5, ϵ ₁=0.3, ϵ ₂=0.3, χ=28, γ=5, k=50); bottom left (c=5, w=22, π ₁=0.02, π ₂=0.004, τ ₁=0.005, τ ₂=0.01, δ ₁=0.001, δ ₂=0.001, κ ₁=10, κ ₂=10, ϵ ₁=3, ϵ ₂=0.2, χ=26.67, γ=0.03, k=45); bottom center (c=5, w=22, π ₁=0.05, π ₂=0.006, τ ₁=0.04, τ ₂=0.015, δ ₁=0.001, δ ₂=0.001, κ ₁=10, κ ₂=10, ϵ ₁=6, ϵ ₂=0.1, χ=3.33, γ=0.03, k=45); bottom right (c=5, w=22, π ₁=0.05, π ₂=0.0056, τ ₁=0.005, τ ₂=0.018, δ ₁=0.001, δ ₂=0.001, κ ₁=5.5, κ ₂=10, ϵ ₁=6, ϵ ₂=0.1, χ=16.17, γ=0.03, k=45)

Characterization of static chemokine fields and dynamics of toy model. The upper left panel shows the static concentration of CXCL12 and CXCL13, and the upper center panel shows the gradient of CXC12 and CXC13. The upper right panel shows the steady state solutions for the receptor density induced by local chemokine concentration, with dashed curves representing chemokine concentrations (blue = CXCL12, red = CXCL13) and solid curves representing steady state receptor densities if a cell was kept fixed at that position (blue = CDCR4, red = CXCR5). The lower panels show the steady state solutions for the velocity v as s=r ₁/r ₂ varies. The black curve shows the rate of change of v, and the red circles indicate stable steady states. In the left panel, s=0 and the system only has a single steady state at x≈1.5. In the middle panel, s=1 and the system is bistable. Finally, in the right panel, s=4.5 and the system has lost the steady state at x≈1.5 and only the steady state at x≈−1.5 remains. Scaled chemokine concentrations represented by f ₁(x) and f ₂(x) are shown as dashed lines for reference (Color figure online)

Comparison of sample oscillations from the 1D phenomenological model (top), IBM simulation with no stochasticity (middle) and IBM simulation with stochasticity (bottom). Top—Sample trajectories for the six (k, π ₁) parameter pairs shown in Fig.  found using a numerical ODE solver. Middle—Sample trajectories for the six (k, π ₁) parameter pairs shown in obtained by running the 3D simulation for a single cell using identical parameter values ″ with σ=0. There is excellent agreement with the ODE solutions shown above. Very minor differences (e.g., slight change in periodicity) are likely due to spatial discretization (5 μm per side voxels with trilinear interpolation) and lower temporal resolution of the 3D simulation compared with the adaptive numerical integrator used in the ODE solution. Bottom—Sample trajectories for the six (k, π ₁) parameter pairs shown in Fig.  obtained by running the 3D simulation for a single cell again with identical parameters except for σ=2. The overall qualitative behavior is similar to that seen with σ=0, but the presence of stochasticity reveals the nearby attractor structure (see sporadic spikes in top row, middle column, and bottom row, last column) reminiscent of stochastic resonance

Bifurcation plots and sample trajectories for the toy model. Left panel shows the 1D bifurcation plot as the separation k between DZ and LZ is varied. Oscillations arise as k is increased above the threshold labeled as H1 where a supercritical Hopf bifurcation occurs. The dotted line segment indicates the parameter range for k where the equilibrium solution is unstable and oscillations exist. At approximately k=39.5, a subcritical Hopf bifurcation occurs at H2. The equilibrium becomes stable again at the saddle node bifurcation LP2 and there are no oscillations beyond this value. It is also possible to go beyond single parameters and identify parameter combinations where bifurcations occur as illustrated by the middle panel that shows the continuation of the Hopf bifurcation as k and the rate of synthesis of CXCR4 (π ₁) are simultaneously varied. The right panel shows sample trajectories for the pairs of (k, π ₁) parameters indicated by crosses in the middle panel, with trajectories for each parameter pair plotted matching the same color cross. Fixed parameters are chosen to be within reasonable biological ranges and to generate oscillations on the correct time scale (hours): c=20, w=15.7, π ₂=1, δ ₁=0.001, δ ₂=0.001, γ=1. In the left panel, π ₁=0.3, while in the center panel, the red (k=40, π ₁=0.35), green (k=45, π ₁=0.58) and blue (k=50, π ₁=0.805) crosses represent (k,π ₁) pairs that have oscillatory behaviors (Color figure online)

Bifurcation plots and sample trajectories for the reduced germinal center model. Left panel shows the 1D bifurcation plot as the separation k between DZ and LZ is varied. Oscillations arise as k is increased above the threshold labeled as H1 where a supercritical Hopf bifurcation occurs. The dotted line segment indicates the parameter range for k where the equilibrium solution is unstable and oscillations exist. At approximately k=52, a subcritical Hopf bifurcation occurs at H2. The equilibrium becomes stable again at the saddle node bifurcation LP2 and there are no oscillations beyond this value. It is also possible to go beyond single parameters and identify parameter combinations where bifurcations occur as illustrated by the right panel that shows the continuation of the Hopf bifurcation as k and the rate of synthesis of CXCR4 (π ₁) are simultaneously varied. Fixed parameters: c=10, w=25, π ₂=0.1, τ ₁=0.06, τ ₂=0.06, δ ₁=0.006, δ ₂=0.006, κ ₁=1, κ ₂=0.1, ϵ ₁=0.3, ϵ ₂=0.3, χ=28, ζ=1, γ=5. In the left panel, π ₁=0.15, while in the right panel, the cyan (k=43, π ₁=0.9), black (k=44, π ₁=0.75), and green (k=50, π ₁=0.15) circles represent (k, π ₁) pairs that have oscillatory behavior, while the magenta (k=44, π ₁=0.9), blue (k=50, π ₁=0.25), and red (k=50, π ₁=0.05) circles represent (k, π ₁) pairs that have steady state solutions. In particular, the green circle (k=50, π ₁=0.15) corresponds to a solution in the unstable equilibrium region of the left panel, where oscillatory behavior is predicted. Sample trajectories corresponding to these parameter pairs are shown in Fig.  (Color figure online)