Vein fate determined by flow-based but time-delayed integration of network architecture

  1. Sophie Marbach
  2. Noah Ziethen
  3. Leonie Bastin
  4. Felix K Bäuerle
  5. Karen Alim  Is a corresponding author
  1. Courant Institute of Mathematical Sciences, New York University, United States
  2. Max Planck Institute for Dynamics and Self-Organization, Germany
  3. Center for Protein Assemblies and Department of Bioscience, School of Natural Sciences, Technical University of Munich, Germany
20 figures, 11 videos, 4 tables and 1 additional file

Figures

Diverse vein dynamics emerge during network reorganization.

(A) Close-up and (B) full network analysis of vein radius dynamics and associated shear rate in P. polycephalum. (i) Bright-field images of reorganizing specimens allow us to record vein dynamics. (i…

Shear rate induces vein adaptation with a time delay.

(A) Principle of the cross-correlation in time between the delayed time-averaged shear rate τ(t-tdelay) and the time-averaged radius change dadt(t). The plot shows the delayed curve with the best score. (B) …

Stable and unstable vein dynamics are predicted within the same model.

(A) Translation of (i) a bright field image of specimen into (ii) vein networks; each vein is modeled as a flow circuit link. (iii) Reduction of (ii) via Northon’s theorem into an equivalent and …

Feedback parameters integrate the network’s architecture and provide information on vein relative location.

Full network maps of the same specimen as in Figure 1B, at the beginning of the observation, of (A) the average fluid pressure in a vein P and (B) of the relative resistance R/Rnet. The fluid pressure P is defined up to an additive constant. Grey veins in (B) correspond to bottleneck veins or dangling ends for which Rnet can not be defined. The color scales indicate the magnitude of each variable in each colored vein. For example, in (B), the red arrow indicates a vein with large relative resistance R/Rnet.

Network architecture controls vein fate as exemplified in three cases.

(A–C) (i) determining factors mapped out from experimental data for the specimen of Figure 1B and (ii) typical trajectories from data. All pink (respectively blue) arrows indicate shrinking …

Avalanche of sequentially vanishing veins.

(A) Time series of network reorganization. Each vein is colored according to the ratio between the resistance of an individual vein R and the rest of the network Rnet in each vein. Red arrows …

Appendix 1—figure 1
Extracting average shear rates from shear rate data.

(A) Time zoom of a close-up data set (that of Figure 1A.iii, #K) showing the doubling of the frequency of the shear rate compared to the radius. (B) Minimal model example with short timescale and …

Appendix 1—figure 2
Vascular adaptation dynamics for all close up experiments, using time-averaged shear rates τ and time-averaged radius a.

The letters indicate the data set names, and are used consistently throughout the manuscript. Typical classification of vein dynamics is indicated for each plot.

Appendix 1—figure 3
Vascular adaptation dynamics for a few veins in the full network #1 using time-averaged shear rates τ and time-averaged radius a.

The veins shown are chosen randomly but distributed throughout the network. The network sketch on the right hand side shows circles indicating at their center the location of each vein, with …

Appendix 1—figure 4
Additional data on the full network specimen #2.

(A - B) Bright field images of a specimen with long time dynamics of vanishing veins, specimen 2. (c) Mean shear rate τ, (D) pressure P and (E) relative resistance R/Rnet at the initial stage. (F) …

Appendix 1—figure 5
Additional data on the full network specimen #3.

(A–B) Bright field images of a specimen with long time dynamics of vanishing veins, specimen . (c) Mean shear rate τ, (D) pressure P and (E) relative resistance R/Rnet at the initial stage. (F) Time …

Appendix 1—figure 6
Cross correlation between average vein radius and different flow-based parameters (A) the connected resistance Rnet, (B) the relative resistance R/Rnet, (C) the vein outflow Q and (E) the local pressure P.

We also present maps of the connected resistance Rnet in (C) and of the vein outflow Q in (F). The color scales indicate the magnitude of each variable in each colored vein. All cross-correlations …

Appendix 2—figure 1
Time delay extracted is independent of averaging technique or oscillation frequency.

(A) Time delays measured without extracting trends from data (same plot as Figure 2C but without extracting trends). (B) Extracting the time delay for model data with (i) model data and extracted …

Appendix 2—figure 2
Time delay statistics from full networks.

Distribution of best time delays for all veins in the network (#1, with about 10,000 vein segments and #2 and #3, both of which have about 30,000 vein segments). (insets). Network maps – not to scale.

Appendix 2—figure 3
Time delays obtained with cross-correlation method on stable close-up data sets.

(Left hand side) Time-averaged shear rate τ (red) and radius change (da/dt) with time for each vein (#E, #F, #G, #K), as well as time delayed shear right producing the best cross correlation τ(t-tdelay). …

Appendix 2—figure 4
Evaluation of all three model parameters tadapt, tdelay and τ0 from suitable data sets (#E, #F, #G, #K).

In the left column the distribution of the obtained time delays using Equation 1 and Equation 2 over a distribution of time windows is depicted. To obtain time windows of approximately constant …

Appendix 2—figure 5
Fit results graphical representation of model Equation 1 and Equation 2 using a fixed time delay of tdelay=120 s for all 12 close-up data sets on a given time window for each data set.

The obtained fit parameters are reported in Appendix 2—table 2. All shear rate and radius data presented is time-averaged.

Appendix 2—figure 6
Fit results of model Equation 1 and Equation 2 for 15 randomly selected veins on specimen #1.

The corresponding fitted parameters are reported in Appendix 2—table 3. The vein positions correspond to those indicated in Appendix 1—figure 3.

Appendix 3—figure 1
Principle of the calculation of Rnet on the basis of a few examples.

The networks are different in all cases except networks D-E-F are the same. D-E-F differ in which vein is under scrutiny. For each case, equivalent resistances Rnet of the rest of the network relative …

Appendix 4—figure 1
All circuits discussed in the text.

(A) General circuit for a vein connected to the rest of the network. (B) Dangling ends; (i) Electric circuit and notations and (ii) stability diagram in the shear radius space. (C) Loops; (i) …

Videos

Video 1
Bright field stacked images of the close-up specimen including veins #H, #I and #J in Figure 1A.i.

Frame sequence: 5 frames at 600 ms and 1 frame at 2.5 s. Scale 0.353 μm/pix.

Video 2
Bright field stacked images of the close-up specimen including vein #K in Figure 1A.ii.

Frame sequence: 5 frames at 600 ms and 1 frame at 5 s. Scale 0.25 μm/pix.

Video 3
Bright field stacked images of the full network specimen #1 in Figure 1B.i.

Frame rate 6 s and scale 5.03 μm/pix.

Appendix 1—video 1
Bright field stacked images of the close-up specimen including vein #A.

Frame sequence: 5 frames at 60 ms and 1 frame at 5 s. Scale 0.574 μm/pix.

Appendix 1—video 2
Bright field stacked images of the close-up specimen including vein #B.

Frame sequence: 5 frames at 60 ms and 1 frame at 2.5 s. Scale 0.48 μm/pix.

Appendix 1—video 3
Bright field stacked images of the close-up specimen including vein #C.

Frame sequence: 5 frames at 60 ms and 1 frame at 2.5 s. Scale 0.408 μm/pix.

Appendix 1—video 4
Bright field stacked images of the close-up specimen including vein #D.

Frame sequence: 5 frames at 600 ms and 1 frame at 2.5 s. Scale 0.25 μm/pix.

Appendix 1—video 5
Bright field stacked images of the close-up specimen including vein #E, #F and #G.

Frame sequence: 5 frames at 60 ms and 1 frame at 2.5 s. Scale 1.06 μm/pix.

Appendix 1—video 6
Bright field stacked images of the close-up specimen including vein #L.

Frame sequence: 5 frames at 600 ms and 1 frame at 2.5 s. Scale 0.513 μm/pix.

Appendix 1—video 7
Bright field stacked images of the full network specimen #2.

Frame rate 6s and scale 5.36 μm/pix.

Appendix 1—video 8
Bright field stacked images of the full network specimen #3.

Frame rate 6 s and scale 12.26 μm/pix.

Tables

Appendix 1—table 1
List of commonly used variables in our work in alphabetical order and significance.

Short length scale variations correspond to variables that can vary strongly from one vein to a neighboring vein, while long length scale variations vary smoothly throughout the network. Variables …

VariableSignificanceLength scale variationsTimescale variations
aRadius of a veinShortShort and long
ϵRelative contraction amplitude
LLength of a vein
μInner fluid viscosityLongLong
ω=2π/TContraction frequency
PInner fluid pressureLongShort and long
P0
Atmospheric pressure
QFluid flow pervading a veinShortShort and long
QinFluid flow generated by peristaltic contractions in a veinShortShort and long
QnetFluid flow generated by the rest of the networkShortShort and long
RResistance of a vein, R=8μL/πa4ShortShort and long
RnetResistance of the rest of the network attached to a vein at both vein endsShortShort and long
τShear rate on a vein’s inner wallShortShort and long
τsSensed shear rate for adaptationShortLong
τ0Steady state shear rate, τ0τactive-P-P0/μLongLong
τactiveActive contribution to τ0, due to energetic consumption from the actomyosin cortex to maintain contractionsLongLong
tTime
tadapt10-100minLong-time adaptation timescaleLongLong
tdelay2minDelay between adaptation and shear rateLongLong
T1-2minPeristaltic contraction period
XLong-time average of XLong
Appendix 2—table 1
Summary of all fitting parameters of sample trajectories depicted in Appendix 2—figure 4, right-hand side.

These fits include fitting of tdelay. The fits are done over a range in time t[tmin,tmax].

Experimenttdelayinstadaptinsτ0ins1tmintmaxinminerrorϵerr
data set G∗69 ± 3903 ± 1905.4 ± 0.0425–461.6 %
data set F∗82 ± 3335 ± 85.7 ± 0.0390–1020.17 %
data set E∗352 ± 7798 ± 33.07 ± 0.0267–1040.6 %
data set K∗76 ± 485 ± 43.63 ± 0.0150–902.7 %
Appendix 2—table 2
Summary of all fitting parameters with a fixed time delay of tdelay=120s for the 12 close-up data sets.

Note that when a data set name is repeated, it corresponds to fitting results over different time ranges t[tmin,tmax] for the same data set.

Experimenttadaptinsτlocins1tmintmaxinminerrorϵerr
data set A2682 ± 8311.48 ± 0.170–170.13 %
data set A220 ± 100.44 ± 0.0017–330.22 %
data set A246 ± 9601.21 ± 0.5967–830.68 %
data set B1195 ± 201.43 ± 0.0010–751.4 %
data set C5948 ± 1220.97 ± 0.000–831.9 %
data set D1531 ± 690.58 ± 0.010–332.0 %
data set D377 ± 60.79 ± 0.008–503.2 %
data set E1312 ± 371.46 ± 0.010–210.93 %
data set E1894 ± 493.05 ± 0.0258–830.66 %
data set F4323 ± 3193.30 ± 0.060–421.9 %
data set F4323 ± 3193.30 ± 0.060–421.9 %
data set F910 ± 245.08 ± 0.0383–1000.27 %
data set G1153 ± 516.07 ± 0.0412–290.38 %
data set G1808 ± 3221.00 ± 0.0942–540.61 %
data set G1233 ± 546.68 ± 0.0783–1000.31 %
data set H355 ± 82.35 ± 0.020–251.4 %
data set I274 ± 52.27 ± 0.010–251.6 %
data set J601 ± 1321.41 ± 0.090–253.1 %
data set K278 ± 63.16 ± 0.008–672.0 %
data set K85 ± 23.70 ± 0.0045–750.6 %
data set K862 ± 1193.40 ± 0.1367–1081.5 %
data set K123 ± 24.18 ± 0.00108–1330.53 %
data set L2553 ± 2480.27 ± 0.016–231.3 %
Appendix 2—table 3
Summary of fitted parameters for a 15 randomly selected veins in the full network Appendix 2—figure 6.

tdelay was established with cross-correlation, while tadapt and τ0 were obtained through linear least-squares fitting. Error bars correspond to the 95% confidence interval and N.R. corresponds to non …

Vein index in full networktdelayinstadaptinsτ0ins1errorϵerr
errorϵerrwithtdelay=0
a241800 ± 2500.76 ± 0.028.1%7.9%
b781520 ± 400.89 ± 0.022.6%2.9%
c3601500 ± 150N.R.1.4%1.4%
d1201800 ± 100N.R.17%23%
e120900 ± 700.2 ± 0.057.4%11%
f1862750 ± 1000.38 ± 0.055.2%6.3%
g422450 ± 500.94 ± 0.035.0%5.1%
h246N.R.0.26 ± 0.038%11%
i186870 ± 400.26 ± 0.013.3%6.9%
j542200 ± 1001.1 ± 0.25.2%5.0%
k288730 ± 504.0 ± 1.04.7%6.7%
l1084050 ± 200N.R.1.1%1.2%
m36010000 ± 30000.62 ± 0.16.1%8.0%
n182050 ± 2000.73 ± 0.023.7%3.8%
o547300 ± 30000.48 ± 0.126.5%6.8%

Additional files

Download links