Figures and data in Vein fate determined by flow-based but time-delayed integration of network architecture

Version of Record: June 1, 2023
Accepted Manuscript: March 14, 2023

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Sophie Marbach

Noah Ziethen

Leonie Bastin

Felix K Bäuerle

Karen Alim

(2023)
Vein fate determined by flow-based but time-delayed integration of network architecture

eLife 12:e78100.

https://doi.org/10.7554/eLife.78100
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Figures
Videos
Tables
Additional files

20 figures, 11 videos, 4 tables and 1 additional file

Figures

Figure 1

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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…

Figure 2

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Shear rate induces vein adaptation with a time delay.

(A) Principle of the cross-correlation in time between the delayed time-averaged shear rate $⟨ τ (t - t_{delay}) ⟩$ and the time-averaged radius change $\frac{d ⟨ a ⟩}{d t} (t)$ . The plot shows the delayed curve with the best score. (B) …

Figure 3

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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 …

Figure 4

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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 $R / R_{net}$ . The fluid pressure $⟨ P ⟩$ is defined up to an additive constant. Grey veins in (B) correspond to bottleneck veins or dangling ends for which $R_{net}$ 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 $R / R_{net}$ .

Figure 5

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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 …

Figure 6

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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 $R_{net}$ in each vein. Red arrows …

Appendix 1—figure 1

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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

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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

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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

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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 $R / R_{net}$ at the initial stage. (F) …

Appendix 1—figure 5

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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 $R / R_{net}$ at the initial stage. (F) Time …

Appendix 1—figure 6

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Cross correlation between average vein radius and different flow-based parameters (A) the connected resistance $R_{net}$ , (B) the relative resistance $R / R_{net}$ , (C) the vein outflow $Q$ and (E) the local pressure $⟨ P ⟩$ .

We also present maps of the connected resistance $R_{net}$ 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

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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

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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

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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 ( $d ⟨ a ⟩ / d t$ ) with time for each vein (#E, #F, #G, #K), as well as time delayed shear right producing the best cross correlation $⟨ τ (t - t_{delay}) ⟩$ . …

Appendix 2—figure 4

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Evaluation of all three model parameters $t_{a d a p t}$ , $t_{d e l a y}$ and $τ_{0}$ 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

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Fit results graphical representation of model Equation 1 and Equation 2 using a fixed time delay of $t_{d e l a y} = 120 s$ 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

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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

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Principle of the calculation of $R_{net}$ 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 $R_{net}$ of the rest of the network relative …

Appendix 4—figure 1

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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

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posterframe for video — 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

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Video 3

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Appendix 1—video 1

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Appendix 1—video 2

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Appendix 1—video 3

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Appendix 1—video 4

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Appendix 1—video 5

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Appendix 1—video 6

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Appendix 1—video 7

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Appendix 1—video 8

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Variable	Significance	Length scale variations	Timescale variations
$a$	Radius of a vein	Short	Short and long
$ϵ$	Relative contraction amplitude	–	–
$L$	Length of a vein	–	–
$μ$	Inner fluid viscosity	Long	Long
$ω = 2 π / T$	Contraction frequency	–	–
$P$	Inner fluid pressure	Long	Short and long
$P_{0}$	Atmospheric pressure	–	–
$Q$	Fluid flow pervading a vein	Short	Short and long
$Q_{in}$	Fluid flow generated by peristaltic contractions in a vein	Short	Short and long
$Q_{net}$	Fluid flow generated by the rest of the network	Short	Short and long
$R$	Resistance of a vein, $R = 8 μ L / π a^{4}$	Short	Short and long
$R_{net}$	Resistance of the rest of the network attached to a vein at both vein ends	Short	Short and long
$τ$	Shear rate on a vein’s inner wall	Short	Short and long
$τ_{s}$	Sensed shear rate for adaptation	Short	Long
$τ_{0}$	Steady state shear rate, $τ_{0} \sim τ_{active} - ⟨ P - P_{0} ⟩ / μ$	Long	Long
$τ_{active}$	Active contribution to $τ_{0}$ , due to energetic consumption from the actomyosin cortex to maintain contractions	Long	Long
$t$	Time	–	–
$t_{adapt} ≃ 10 - 100 \min$	Long-time adaptation timescale	Long	Long
$t_{delay} ≃ 2 \min$	Delay between adaptation and shear rate	Long	Long
$T ≃ 1 - 2 \min$	Peristaltic contraction period	–	–
$⟨ X ⟩$	Long-time average of $X$	–	Long

Experiment	$t_{d e l a y} i n s$	$t_{a d a p t} i n s$	$τ_{0} i n s^{- 1}$	$t_{m i n} - t_{m a x} i n m i n$	$e r r o r ϵ_{e r r}$
data set G∗	69 ± 3	903 ± 190	5.4 ± 0.04	25–46	1.6 %
data set F∗	82 ± 3	335 ± 8	5.7 ± 0.03	90–102	0.17 %
data set E∗	352 ± 7	798 ± 3	3.07 ± 0.02	67–104	0.6 %
data set K∗	76 ± 4	85 ± 4	3.63 ± 0.01	50–90	2.7 %

Experiment	$t_{a d a p t} i n s$	$τ_{loc} i n s^{- 1}$	$t_{m i n} - t_{m a x} i n m i n$	$e r r o r ϵ_{e r r}$
data set A	2682 ± 831	1.48 ± 0.17	0–17	0.13 %
data set A	220 ± 10	0.44 ± 0.00	17–33	0.22 %
data set A	246 ± 960	1.21 ± 0.59	67–83	0.68 %
data set B	1195 ± 20	1.43 ± 0.00	10–75	1.4 %
data set C	5948 ± 122	0.97 ± 0.00	0–83	1.9 %
data set D	1531 ± 69	0.58 ± 0.01	0–33	2.0 %
data set D	377 ± 6	0.79 ± 0.00	8–50	3.2 %
data set E	1312 ± 37	1.46 ± 0.01	0–21	0.93 %
data set E	1894 ± 49	3.05 ± 0.02	58–83	0.66 %
data set F	4323 ± 319	3.30 ± 0.06	0–42	1.9 %
data set F	4323 ± 319	3.30 ± 0.06	0–42	1.9 %
data set F	910 ± 24	5.08 ± 0.03	83–100	0.27 %
data set G	1153 ± 51	6.07 ± 0.04	12–29	0.38 %
data set G	1808 ± 322	1.00 ± 0.09	42–54	0.61 %
data set G	1233 ± 54	6.68 ± 0.07	83–100	0.31 %
data set H	355 ± 8	2.35 ± 0.02	0–25	1.4 %
data set I	274 ± 5	2.27 ± 0.01	0–25	1.6 %
data set J	601 ± 132	1.41 ± 0.09	0–25	3.1 %
data set K	278 ± 6	3.16 ± 0.00	8–67	2.0 %
data set K	85 ± 2	3.70 ± 0.00	45–75	0.6 %
data set K	862 ± 119	3.40 ± 0.13	67–108	1.5 %
data set K	123 ± 2	4.18 ± 0.00	108–133	0.53 %
data set L	2553 ± 248	0.27 ± 0.01	6–23	1.3 %

$t_{d e l a y}$ was established with cross-correlation, while $t_{a d a p t}$ and $τ_{0}$ 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 network	$t_{d e l a y} i n s$	$t_{a d a p t} i n s$	$τ_{0} i n s^{- 1}$	$e r r o r ϵ_{e r r}$	$e r r o r ϵ_{e r r} w i t h t_{d e l a y} = 0$
a	24	1800 ± 250	0.76 ± 0.02	8.1%	7.9%
b	78	1520 ± 40	0.89 ± 0.02	2.6%	2.9%
c	360	1500 ± 150	N.R.	1.4%	1.4%
d	120	1800 ± 100	N.R.	17%	23%
e	120	900 ± 70	0.2 ± 0.05	7.4%	11%
f	186	2750 ± 100	0.38 ± 0.05	5.2%	6.3%
g	42	2450 ± 50	0.94 ± 0.03	5.0%	5.1%
h	246	N.R.	0.26 ± 0.03	8%	11%
i	186	870 ± 40	0.26 ± 0.01	3.3%	6.9%
j	54	2200 ± 100	1.1 ± 0.2	5.2%	5.0%
k	288	730 ± 50	4.0 ± 1.0	4.7%	6.7%
l	108	4050 ± 200	N.R.	1.1%	1.2%
m	360	10000 ± 3000	0.62 ± 0.1	6.1%	8.0%
n	18	2050 ± 200	0.73 ± 0.02	3.7%	3.8%
o	54	7300 ± 3000	0.48 ± 0.12	6.5%	6.8%

Additional files

MDAR checklist: https://cdn.elifesciences.org/articles/78100/elife-78100-mdarchecklist1-v2.docx
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Cite this article

Diverse vein dynamics emerge during network reorganization.

Shear rate induces vein adaptation with a time delay.

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

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

Network architecture controls vein fate as exemplified in three cases.

Avalanche of sequentially vanishing veins.

Extracting average shear rates from shear rate data.

Additional data on the full network specimen #2.

Additional data on the full network specimen #3.

Time delay extracted is independent of averaging technique or oscillation frequency.

Time delay statistics from full networks.

Time delays obtained with cross-correlation method on stable close-up data sets.

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

Principle of the calculation of Rnet<math id="inf450"><msub><mi>R</mi><mi>net</mi></msub></math> on the basis of a few examples.

All circuits discussed in the text.

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

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

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

Bright field stacked images of the close-up specimen including vein #A.

Bright field stacked images of the close-up specimen including vein #B.

Bright field stacked images of the close-up specimen including vein #C.

Bright field stacked images of the close-up specimen including vein #D.

Bright field stacked images of the close-up specimen including vein #E, #F and #G.

Bright field stacked images of the close-up specimen including vein #L.

Bright field stacked images of the full network specimen #2.

Bright field stacked images of the full network specimen #3.

List of commonly used variables in our work in alphabetical order and significance.

Summary of all fitting parameters of sample trajectories depicted in Appendix 2—figure 4, right-hand side.

Summary of all fitting parameters with a fixed time delay of tdelay=120s<math id="inf428"><mrow><msub><mi>t</mi><mi>delay</mi></msub><mo>=</mo><mrow><mn>120</mn><mtext></mtext><mi mathvariant="normal">s</mi></mrow></mrow></math> for the 12 close-up data sets.

Summary of fitted parameters for a 15 randomly selected veins in the full network Appendix 2—figure 6.

MDAR checklist

Principle of the calculation of $R_{net}$ on the basis of a few examples.

Summary of all fitting parameters with a fixed time delay of $t_{delay} = 120 s$ for the 12 close-up data sets.