Synchronization and Random Triggering of Lymphatic Vessel Contractions

Abstract

The lymphatic system is responsible for transporting interstitial fluid back to the bloodstream, but unlike the cardiovascular system, lacks a centralized pump-the heart-to drive flow. Instead, each collecting lymphatic vessel can individually contract and dilate producing unidirectional flow enforced by intraluminal check valves. Due to the large number and spatial distribution of such pumps, high-level coordination would be unwieldy. This leads to the question of how each segment of lymphatic vessel responds to local signals that can contribute to the coordination of pumping on a network basis. Beginning with elementary fluid mechanics and known cellular behaviors, we show that two complementary oscillators emerge from i) mechanical stretch with calcium ion transport and ii) fluid shear stress induced nitric oxide production (NO). Using numerical simulation and linear stability analysis we show that the newly identified shear-NO oscillator shares similarities with the well-known Van der Pol oscillator, but has unique characteristics. Depending on the operating conditions, the shear-NO process may i) be inherently stable, ii) oscillate spontaneously in response to random disturbances or iii) synchronize with weak periodic stimuli. When the complementary shear-driven and stretch-driven oscillators interact, either may dominate, producing a rich family of behaviors similar to those observed in vivo.

Fig 1. A single lymphangion while opening and closing with valves at the inlet and outlet.

Fig 2. Stretch-Ca dynamics at constant NO with noise triggering.

At lower pressure a-c) p_m = 100 Pa, t_Ca = 1s and t_mech = 1.22 s, and at a higher pressure d-f)p_m = 500Pa, t_Ca = 1s, and t_mech = 0.24 s. The noise level is σ = 0.01. In each case the overall time constant is given approximately by the greater of t_Ca and t_mech. The period is predicted approximately by t_c log(C_max/σ) where C_max is the amplitude of the change in Ca²⁺ concentration.

Fig 3. Synchronization of stretch-Ca dynamics with small amplitude sinusoidal inputs at constant NO.

At higher pressure p_m = 500 Pa, t_Ca = 1 s, and t_mech = 0.24 s with input amplitude of 0.01 and input frequencies of a) 0.3 Hz, b) 0.2 Hz, c) 0.1 Hz and d) 0.05 Hz. The tick marks indicate the beginnings of successive cycles of the sinusoidal input signal. The ratio of output frequency to input frequency is at the right of each panel.

Fig 4. Location of the shear-NO eigenvalues in the complex plane as the baseline radius R₁ increases parametrically.

The arrows indicate the direction of increasing R₁. The eigenvalues during contraction always have a negative real part indicating stability. When the baseline radius is large enough to move the eigenvalues beyond point B the system is inherently stable during dilation. At point B the system is marginally stable and will oscillate at frequency f = (E₁K_NO/D)^1/2/2π. For smaller baseline radius, between points A and B, the response is unstable, but oscillatory. And when the baseline radius is smaller at point A, the dynamic component of the radius increases exponentially without oscillation until the radius is large enough to reach the range between A and B where oscillations can occur.

Fig 5. a) Generic time response and b) phase portrait of the shear-NO oscillator.

Following a small perturbation shown as a green arrow near equilibrium at point 0, by either a decrease in radius or an increase in concentration, the radius increases unstably (red arrows $\dot{R}>0$) until $\dot{R}=0$ at point 2. Thereafter, the trajectory begins a stable return (blue arrows $\dot{R}<0$) to equilibrium at point 0. NO reaches its peak concentration at point 1 before the radius reaches its maximum, but this point does not directly influence the stability of the system.

Fig 6. Shear-NO dynamics at constant Ca²⁺ with noise triggering.

At low pressure a-c) p_m = 100 Pa, t_Ca = 1 s, t_mech = 1.22 s, t_NO = 0.2 s and t_FNO = 3.35 s where the shear-NO oscillator is unstable (t_mech + t_NO < t_FNO) and at a higher pressure d-f) p_m = 500 Pa, t_Ca = 1 s, t_mech = 0.24 s, t_NO = 0.2 s and t_FNO = 0.21 s where the shear-NO oscillator is stable (t_mech + t_NO > t_FNO) yielding little change in radius. The noise level is σ = 0.01. In each case the overall time constant for return to equilibrium is given approximately by t_mech + t_NO + t_FNO. The period is predicted approximately by t_c log(C_max/σ) where C_max is the amplitude of the changes in NO concentration.

Fig 7. Synchronization of shear-NO dynamics with small amplitude sinusoidal inputs at constant Ca.

At lower pressure p_m = 100 Pa, t_Ca = 1 s, t_mech = 1.22 s, t_NO = 0.2 s and t_FNO = 3.35 s where the shear-NO oscillator is unstable (t_mech + t_NO < t_FNO) with input amplitude of 0.01 and input frequencies of a) 0.08 Hz, b) 0.04 Hz, c) 0.02 Hz and d) 0.01 Hz. The tick marks indicate the beginnings of successive cycles of the sinusoidal input signal. The ratio of the output frequency to the input frequency is at the right of each panel.

Fig 8. Synchronization of the shear-NO oscillator with a sinusoidal input.

a) Color bands indicate domains in which the output and input frequencies lock into simple, integer ratios. The red curve gives an estimate of the autonomous frequency from f = 1/(t_clog(C_NOmax/σ) where the amplitude of the noise has been replaced by the amplitude of the input sinusoid, b) Output frequencies at input amplitude of 0.01 showing a so-called Devil's staircase of discrete, rational values.

Fig 9. Fully coupled Ca²⁺ and NO dynamics operating autonomously.

a-c) Low pressure (small radius) with overshoot of the nominal radius due to instability in the NO dynamics leading to Fig 8 trajectories in phase space (t_mech + t_NO < t_FNO) p_m = 100 Pa, t_Ca = 1 s, t_mech = 1.22 s, t_NO = 0.2 s, and t_FNO = 3.35 s, d-f) Higher pressure (large radius) without overshoot due to stable NO dynamics (t_mech + t_NO > t_FNO) p_m = 500 Pa, t_Ca = 1 s, t_mech = 0.24 s, t_NO = 0.2 s and t_FNO = 0.21 s. Note the decaying oscillations of the NO concentration in panel d. Near marginal stability (t_mech + t_NO + t_FNO) the NO concentrations oscillates at a frequency approximated by f = (E₁K_NO/D)^1/2/2π which corresponds to where the eigenvalues cross the imaginary axis at point B on Fig 4.

Fig 10. Typical in vivo experimental measurements of lymphangion diameter (a,c) in a mouse with phase portraits of diameter vs. rate of diameter change (b,d) [20].

a,b) Wild type mice with Ca²⁺ and NO active. c,d) NO effects genetically deleted in eNOS^-/- mice. Spacing of data points indicates the rate of motion in the phase plane (sampling period 0.21 s). The box surrounds a region of closely spaced points indicative of an equilibrium condition. Orange arrows show the trajectory during a contraction, while green arrows show a dilation. Contraction and dilation dynamics are generally more erratic when the vessel is small and has active NO than when the vessel is larger and has suppressed NO activity.