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

Veins in vascular networks, such as in blood vasculature or leaf networks, continuously reorganize, grow or shrink, to minimize energy dissipation. Flow shear stress on vein walls has been set forth as the local driver for a vein’s continuous adaptation. Yet, shear feedback alone cannot account for the observed diversity of vein dynamics – a puzzle made harder by scarce spatiotemporal data. Here, we resolve network-wide vein dynamics and shear rate during spontaneous reorganization in the prototypical vascular networks of Physarum polycephalum. Our experiments reveal a plethora of vein dynamics (stable, growing, shrinking) where the role of shear is ambiguous. Quantitative analysis of our data reveals that (a) shear rate indeed feeds back on vein radius, yet, with a time delay of 1–3 min. Further, we reconcile the experimentally observed disparate vein fates by developing a model for vein adaptation within a network and accounting for the observed time delay. The model reveals that (b) vein fate is determined by parameters – local pressure or relative vein resistance – which integrate the entire network’s architecture, as they result from global conservation of fluid volume. Finally, we observe avalanches of network reorganization events that cause entire clusters of veins to vanish. Such avalanches are consistent with network architecture integrating parameters governing vein fate as vein connections continuously change. As the network architecture integrating parameters intrinsically arise from laminar fluid flow in veins, we expect our findings to play a role across flow-based vascular networks.

allows us to directly measure radius dynamics a(t) and velocity profiles v(r, t) inside vein segments using particle image velocimetry ( Figure 1A. ii), where t is time and r is the radial coordinate along the tube (all variable names are reported in Appendix 1-table 1). From velocity profiles, we extract the flow rate across a vein's cross-section Q(t) = 2π´v(r, t)rdr . In full networks ( Figure 1B.i, see also Video 3), radius dynamics a(t) are measured for each vein segment and flow rates Q(t) are subsequently calculated numerically integrating conservation of fluid volume via Kirchhoff laws, see Appendix 1. Our imaging techniques resolve vein adaptation over a wide range of vein radii, a = 5 − 70µm . Radii data show rhythmic peristaltic contractions, with a period of T ≃ 1 − 2 min (light blue in Figure 1iii). We calculate shear rate from fluid flows as τ = 4 π |Q| a 3 . Unlike shear stress, shear rate measurements do not require knowledge of the fluid's viscosity and are, therefore, more precise. Since both quantities are directly proportional, the conclusions we draw for shear rate apply to shear stress on the typical timescale of our experiments, where potential aging affects altering fluid viscosity can be neglected. We observe that shear rate τ oscillates with twice the contraction frequency (light red in Figure 1iii). In fact, since flows Q reverse periodically, they oscillate around 0. In the shear rate τ , oscillation periods are even doubled due to taking the absolute value of Q in calculating τ ; see also Appendix 1-figure 1.
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. https://elifesciences.org/articles/78100/figures#video1 To access the long-time behavior of veins, we average out short timescales on the order of T ≃ 1 − 2 min corresponding to the peristaltic contractions (Isenberg and Wohlfarth-Bottermann, 1976). We, thus, focus on the dynamics of the time-averaged radius ⟨a⟩ and shear rate ⟨τ ⟩ on longer timescales, from 10-60 min (full lines in Figure 1iii), corresponding to growth or disassembly of the vein wall, linked to e.g actin fiber rearrangements (Salbreux et al., 2012;Fischer-Friedrich et al., 2016).

Diverse and reproducible vein dynamics
We relate time-averaged shear rate to timeaveraged vein radius and find diverse, complex, yet reproducible trajectories ( Figure 1A, B.iv, see also Appendix 1- figure 4 and Appendix 1figure 5 for additional datasets). To illustrate this diversity, out of 200 randomly chosen veins in the full network of Figure 1B, we find 80 shrinking veins, 100 stable veins, and 20 are not classifiable. In shrinking veins, the relation between shear rate and vein adaption is particularly ambiguous. As the radius of a vein shrinks, the shear rate either monotonically decreases (pink b in Figure 1B. iv), or, monotonically increases (pink d), or, increases at first and decreases again (pink c). For the specimen of Figure 1B, out of the 80 shrinking veins, monotonic decrease is observed for 25%, monotonic increase for 40%, and non-monotonic trajectories 15% of the time. The remaining 20% of vanishing veins are unclassifiable, as their recorded trajectories are too short to allow for any classification. Out of the 12 close-up veins investigated, 4 shrink and vanish, either with monotonic or non-monotonic dynamics (see also Appendix 1-figure 2).
In contrast, stable veins have a specific shear rate-radius relation: usually, stable veins perform looping trajectories in the shear rate-radius space (blue arrows in Figure 1A, B.iv). In the full network, these loops circle clockwise for 80% of 100 observed stable veins. Out of the 12 close-up veins investigated, 6 veins show stable clockwise feedback, 1 shows stable anticlockwise feedback, and 1 is not classifiable. Clockwise circling corresponds to an in/decrease in shear rate followed by an in/ decrease in vein radius, thus, hinting at a shear rate feedback on local vein adaptation. This establishes a potential causality link between shear rate changes and vascular adaptation. In addition, the circular shape of stable vein trajectories suggests that there is a time delay between changes in shear rate and subsequent vein radius changes.
Shear rate and resistance feedback alone can not account for the diversity of vein fates We further test this potential causality link between shear rate and vein adaptation. Based on previous theoretical works (Taber, 1998a;Hacking et al., magnitude of shear rate directly determines vein fate, that is lower shear rate results in a shrinking vein. Yet, this is not corroborated by our experimental measurements. First, despite displaying comparable shear rate and vein radii at the beginning of our data acquisition, some veins are stable (blue a in Figure 1A, B.iv), while others vanish (pink b). We, thus, map out shear rate throughout the entire network at the beginning of our observation, see Figure 1B. ii. We observe that dangling ends have low shear rate, due to flow arresting at the very end of the vein (dark purple terminal veins). Yet, some dangling ends will grow (i.e red dot in Figure 1B.i), in contradiction again with the assumption that 'low shear results in a shrinking vein'. Finally, small veins located in the middle of the organism show high shear rate, yet, will vanish (yellow arrow in Figure 1B. ii, other examples in Appendix 1- figure  4C and Appendix 1- figure 5C). Therefore, the hypothesis that veins with low shear rate should vanish, as they cannot sustain the mechanical effort (Koller et al., 1993;Hoefer et al., 2013), cannot be reconciled with our data. Finally, also other purely geometrical vein characteristics such as vein resistance (Baumgarten and Hauser, 2013), R = 8µL π⟨a⟩ 4 , where µ is the fluid viscosity and L the vein length (Happel and Brenner, 2012), clearly do not determine vein fate either. In fact, geometrical vein characteristics are directly related to vein radius, thus in contradiction with our observation that veins with similar radius can experience different fates (stable blue a in Figure 1iv and vanishing pink b). Therefore, additional feedback parameters must play a role.

Shear rate feedback on individual vein dynamics occurs with a time delay
The link between shear rate feedback and vein adaptation is clearly ambiguous in our data. To understand the feedback mechanism, we now turn to modeling and in-depth analysis.
Vein radius adaptation in response to shear rate Current theoretical models (Hacking et al., 1996;Hu and Cai, 2013;Taber, 1998a;Ronellenfitsch and Katifori, 2016) motivated by Murray's phenomenological rule of minimizing dissipation (Murray, 1926) suggest that vascular adaptation, d⟨a⟩ dt , that is the change in time of the vein radius ⟨a⟩ , is related to shear rate ⟨τ ⟩ via Here, τs(⟨τ ⟩) is the shear rate sensed by a vein wall and is directly related to fluid shear rate ⟨τ ⟩ , in a way that we specify in the following paragraph. The parameter t adapt is the adaptation time to grow or disassemble vein walls corresponding to fiber rearrangement (Salbreux et al., 2012;Fischer-Friedrich et al., 2016) and τ 0 the vein's reference shear rate, corresponding to a steady state regime τs = τ 0 with constant shear rate -in agreement with Murray's law (see Appendix 2.1; Murray, 1926). t adapt and τ 0 are independent variables, constants over the timescale of a vein's adaptation, and could a priori vary from vein to vein, though existing models assume they do not (Hacking et al., 1996;Hu and Cai, 2013;Taber, 1998a;Ronellenfitsch and Katifori, 2016). We here already incorporated two adaptations for our experimental system. First, we specifically indicate with d⟨a⟩ dt that we are interested in vascular adaptation, that is on long-time changes in the vein radius. In contrast, the short timescale variations d(a−⟨a⟩) dt in P. polycephalum are driven by peristaltic contractions (Isenberg and Wohlfarth-Bottermann, 1976) and are not relevant for long-time adaptation. Second, we here, in contrast to all previous work, allow vein radii dynamics to potentially depend via a time delay on the shear rate, by describing radii dynamics as a function of a sensed shear rate, τs(⟨τ ⟩) , which itself depends on the average shear rate ⟨τ ⟩ . We will specify this dependence in the section ''Model with a time delay quantitatively reproduces the data''.
Theoretical models differ in the precise functional dependence on shear rate on the right-hand side of Equation 1, but agree in all using a smooth function f(τs) . We here employ a functional form with a quadratic scaling of the right-hand side on the shear rate f(τs) ∝ τ 2 s that we obtained via a bottom-up derivation from force balance on a vein wall segment in a companion work (Marbach et al., 2023). Within the force balance derivation, the cross-linked actin fiber cortex composing the vein wall responds with a force in the normal direction compared to tangential shear and, hence, drives veins to dilate or shrink in response to shear (Gardel et al., 2008;Janmey et al., 2007; see Appendix 2.1). Experimental data measuring this anisotropic response of fibers in Janmey et al., 2007;Vahabi et al., 2018;Kang et al., 2009 suggest a quadratic dependence of the change in fibers thickness on the applied shear. This quadratic dependence is also consistent with the top-down phenomenological result of Hu et al., 2012. That said, our upcoming results are robust against the specific choice of f(τs) , as long as f increases with |τs| and their exists a non-zero value of shear rate τ 0 corresponding to Murray's steady-state, that is such that f(τ 0 ) = 0 .
Regarding the interpretation of the sensed shear rate τs , it is apparent from our data that the link between shear rate and radius adaptation is not immediate but occurs with a time delay. Figure 1iii indeed shows lag times between peaks in time-averaged shear rate and radius dynamics, ranging from 1 min to 10 min. As a result, τs could correspond to a delayed shear rate compared to the actual one ⟨τ ⟩ . We turn to confirm this assumption and analyze this time delay further.

Statistical analysis of the time delay between shear rate and radius dynamics
We systematically investigate the time delay between shear rate ⟨τ ⟩ and vein adaptation d⟨a⟩ dt . For each vein segment, we calculate the cross-correlation between averaged shear rate ⟨τ ⟩(t − t delay ) and vein adaptation d⟨a⟩ dt (t) as a function of the delay t delay (Figure 2A). Then, we record the value of t delay that corresponds to a maximum ( Figure 2B). Time delays are recorded if the maximum is significant only, that is if the cross-correlation is high enough, and here we choose the threshold to be 0.5. Note, that slight changes in the threshold do not affect our results significantly. Both positive and negative time delays are recorded. Each full network data set contains more than 10,000 vein segments, which allows us to obtain statistically relevant data of t delay ( Figure 2C and see also Appendix 2-figure 2). We present additional methods to extract the time delay also in close-up networks in Appendix 2. Note that t delay is different from t adapt . Although both timescales are relevant to describe adaptation in our specimen: t delay represents the time to sense shear rate signals in vein walls; t adapt represents the time to grow or disassemble a vein wall.
Overall, we find 15 times more veins with positive time delays than with negative time delays for the specimen of Figure 1B (full time delay distribution in Appendix 2- figure 2). This clearly establishes a causality link between shear rate magnitude and radius adaptation. We also find that time delays of 1 to 3 min are quite common with an average of t delay ≃2 min ( Figure 2C). We repeat the analysis over different full network specimens (Appendix 2-figure 2) and close-up veins (Appendix 2-figure 3) and find similar results.
While unraveling the exact biophysical origin of the time delay is beyond the scope of this work, it is important to discuss potential mechanisms. First, the typical time delay measured t delay ≃ 2 min appears close to the contraction period T ≃ 1 − 2 min . This is not an artifact of the analysis (see benchmark test in Appendix 2). Rather, it hints that the cross-linked actomyosin and contractile cortex are key players in the delay. Measured data on the contractile response of cross-linked fibers (Gardel et al., 2008;Janmey et al., 2007;Vahabi et al., 2018) exhibits a time delay of about 1-30 s for in vitro gels. This time delay could accumulate in much longer time delays in vivo (Armon et al., 2018), as is the case in our sample, and potentially reach a time delay of about 2 min . Other mechanical delays could originate from the cross-linked actomyosin gel. For example, the turnover time for actin filaments in living cells ranges from 10 s to 30 s (Fritzsche et al., 2013;Livne et al., 2014;Colombelli et al., 2009), while the viscoelastic relaxation time is 100 s (Joanny and Prost, 2009), both timescales close to our measured time delay.

Model with a time delay quantitatively reproduces the data
Having clearly established the existence of a positive time delay for shear rate feedback on vein adaptation, we must radically deviate from existing models (Hacking et al., 1996;Taber, 1998b;Taber, 1998a;Pries et al., 2005;Secomb et al., 2013) by incorporating the measured time delay t delay explicitly between the shear rate sensed by a vein wall τs and fluid shear rate ⟨τ ⟩ . To this end, we use the phenomenological first-order equation At steady state, we recover a constant shear rate ⟨τ ⟩ = τs = τ 0 , corresponding to Murray's law (Appendix 2) (Murray, 1926). We further verify that our model with the adaptation rule Equation 1 and the time delay shear rate sensing Equation 2 quantitatively accounts for the observed dynamics with physiologically relevant parameters. We fit our 12 close-up data sets, as well as 15 randomly chosen veins of the full network in Figure 1B. We take shear rate data ⟨τ ⟩(t) as input and fit model constants t adapt and τ 0 to reproduce radius data ⟨a⟩ (t) . Note, that t adapt and τ 0 are independent variables that vary from vein to vein, and over long timescales and between specimen (Swaminathan et al., 1997;Puchkov, 2013;Fessel et al., 2017;Lewis et al., 2015;Marbach et al., 2023). To test the robustness of model fits, we employ different strategies to set the time delay t delay before fitting. The time delay is either set to the same average value for all veins, or to the best cross-correlation value for a specific vein, or fitted with a different value for each vein, with no significant change in the resulting goodness of fit and fit parameter values.
Overall, we find a remarkable agreement between fit and data (see example in Figure 2D and Appendix 2 for more results). We find a small relative error on fitted results ϵerr =´dt |⟨a⟩−⟨a⟩ fit | ⟨a⟩ ≃ 0.001 − 0.17 . This suggests that the minimal ingredients of this model are sufficient to reproduce experimental data. Fits without the time delay yield systematically worse results, with larger fitting errors ϵerr (see Figure 2D, dotted black line and Appendix 2-table 3). In all samples, fitting parameters resulted in physically reasonable values. We found t adapt ≃ 10 − 100 min corresponding to long timescale adaptation of vein radii. Note again, the physical difference between the time to adapt vein radius t adapt and the time delay to sense shear rate t delay also translates to orders of magnitude differences with t delay ≃ 2 min and t adapt ≃ 10 − 100 min . This 10-100 min is indeed the timescale over which we observe significant adaptation. Reorganization of biological matter occurs on similar timescales in other comparable systems, from 15 min for individual cells to several days for blood vasculature (Livne et al., 2014;Landau et al., 2018).
When examining fit results of the target shear rate τ 0 it is a priori hard to estimate which values to expect since τ 0 is only reached at steady state. Yet, in our continuously evolving specimen, we never reach steady state and, hence, can not measure τ 0 . However, we can compare τ 0 to shear rate values measured in our specimen and find that they are consistently of the same order of magnitude. Finally, we find that our model yields better results if we fit the data over intermediate time frames (15 min to 40 min), exceeding results of fitting over longer time frames (40 min to 100 min). This is in line with our theoretical expectation (Marbach et al., 2023) that t adapt and τ 0 change over long timescales, since they depend on physical parameters that also change over long timescales, in particular in response to network architecture changes. Since veins typically vanish over 15 min to 40 min and, hence, significant network changes occur over exactly that timescale, t adapt and τ 0 are no longer constant for time frames ≳ 40 min .
While we have focused so far on timescales of individual vein adaptation, we now aim to understand how their individual disparate fates arise. We will show that the origin of different fates resides in the evolution of the rest of the network.
Relative resistance and pressure determine vein fate within a network Stable and unstable vein dynamics are predicted within the same model To capture the impact of the entire network on the dynamics of a single vein modeled by Equations 1; 2, we must specify the flow-driven shear rate ⟨τ ⟩ . Since ⟨τ ⟩ = 4|Q| π⟨a⟩ 3 , it is sufficient to specify the flow rate Q in a vein. Q is coupled to the flows throughout the network by conservation of fluid volume through Kirchhoff's laws, and is, therefore, an indirect measure of network architecture.
We, here, consider the most common vein topology of a vein connected at both ends to the remaining network, more specialized topologies follow in ''Specific vein fates''. The network is then represented by a vein of equivalent resistance Rnet parallel to the single vein of R = 8µL π⟨a⟩ 4 considered within an equivalent flow circuit, see Figure 3A. Rnet is the equivalent resistance corresponding to all the resistances making up the rest of the network, obtained with Kirchhoff's laws (see examples in Appendix 3). Rnet is therefore integrating the network's architecture. Such a reduction of a flow network to a simple equivalent flow circuit is always possible due to Norton's theorem (Morris, 1978).

The time-averaged net flow generated by the vein contractions is
where ϵ is the relative contraction amplitude. The absolute values in this definition are used to measure the net flow. Q in thus measures the mass exchanges between the network and the vein. As mass is conserved, this results in an inflow of Qnet = −Q in , into the rest of the network. Within the vein, a total flow rate Q circulates -see Figure 3A. ii. The flow rate Q through the vein follows from Kirchhoff's second law: QR = −(Q + Qnet)Rnet . We, thus, obtain that the time-averaged shear rate in the vein is The coupled dynamics of {⟨τ ⟩, τs, ⟨a⟩} are now fully specified through Equations 1-3. To simplify our analysis, we now explore the reduced system {τs, ⟨a⟩} by replacing ⟨τ ⟩ in Equation 2 by its expression in Equation 3. Using standard tools of dynamical systems theory, see Appendix 3.3, we now characterize the typical trajectories predicted within the model. Our dynamic system {τs, ⟨a⟩} reproduces the key features of the trajectories observed experimentally. We find two stable fixed points at (0, 0) and (τ 0 , ⟨a⟩ stable (R/Rnet, τ 0 )) , and one unstable fixed point at (τ 0 , ⟨a⟩ unstable (R/Rnet, τ 0 )) (see Figure 3B). The stable fixed point with finite radius, (τ 0 , ⟨a⟩ stable ) corresponds to Murray's steady state. The set of fixed points was also found in a related theoretical study that investigates a phenomenological model resembling Equation 1, yet without any time delay, and examining the stability of a vein, or resistance, connected to a pressure source and another resistance (Hacking et al., 1996). This suggests that the presence of the three fixed points is universal. Furthermore, we find similar dynamical trajectories in the {τs, ⟨a⟩} as those observed experimentally. Trajectories spiral in the clockwise direction near the stable fixed point (τ 0 , ⟨a⟩ stable ) (blue in Figure 3B) and veins shrink with monotonic (dark pink in Figure 3B) or with non-monotonic shear rate decrease (light pink Figure 3B). The dynamics of ⟨τ ⟩ are then closely related to that of τs .

Relative resistance and pressure control vein fate
Analysis of the vein network model as a dynamic system, Equations 1-3, clearly highlights that different vein fates may occur depending on the value of the relative resistance R/Rnet and on the value of the target shear rate τ 0 for that specific vein. We will, therefore, now investigate their values throughout the network more carefully.
Before proceeding, we must specify the meaning of the target shear rate τ 0 . The force balance derivation in Marbach et al., 2023 finds that the shear rate reference τ 0 is related to the local fluid pressure P , as τ 0 ∼ τ active − ⟨P − P 0 ⟩/µ (see short derivation in Appendix 2). Here, P − P 0 characterizes the pressure imbalance between the fluid pressure inside the vein, P , and the pressure outside, P 0 , namely the atmospheric pressure. We recall that µ is the fluid viscosity. Finally, τ active = σ active /µ is a shear rate related to the active stress σ active generated by the actomyosin cortex (Radszuweit et al., 2013;Alonso et al., 2017). The active stress sustains the contractile activity of the vein, and is, therefore, an indirect measure of the metabolic or energetic consumption in the vein. The local pressure P results from solving Kirchhoff's law throughout the network. It is, therefore, indirectly integrating the entire network's morphology. Hence, not only R/Rnet but also τ 0 is a flow-based parameter, integrating network architecture.
In our experimental full network samples, we can calculate both the relative resistance R/Rnet and the local pressure P , and its short time-averaged counterpart ⟨P⟩ , up to an additive constant (see Figure 4). We find that pressure maps of ⟨P⟩ are mostly uniform, except towards dangling ends where relevant differences are observed ( Figure 4A). Hence, particularly in dangling ends, veins with similar shear rate τ may suffer different fates, as described through Equation 1. This is a radical shift compared to previous theoretical works which consider that τ 0 is a constant throughout the network (Taber, 1998a;Hacking et al., 1996;Hu et al., 2012;Secomb et al., 2013;Pries et al., 1998;Pries et al., 2005;Hu and Cai, 2013;Tero et al., 2007).
The relative resistance R/Rnet varies over orders of magnitude ( Figure 4B), with values that are not correlated with vein size (see Appendix 1- figure 6). Rather, R/Rnet indicates how a vein is localized within the network compared to large veins that have lower flow resistance and that serve as highways for transport. For example, a small vein immediately connected to a highway will show a large value of R/Rnet . In this case among all possible flow paths that connect the vein's endpoints, there exists a flow path that consists only of highways, and therefore we expect R ≫ Rnet (see red arrow in Figure 4B). In contrast, a similarly small vein yet localized in between other small veins, further away from highways, will show a smaller value of R/Rnet . In this latter case, all flow paths have to pass through small nearby veins and, hence, have high resistance Rnet ≫ R (see blue arrow in Figure 4B). R/Rnet , therefore, reflects the relative cost to transport fluid through an individual vein rather than through the rest of the network.
The relative resistance R/Rnet is, thus, a natural candidate to account for individual vein adaptation: it measures the energy dissipated by flowing fluid through an individual vein, Q 2 R/2 , compared to rerouting this flow through the rest of the network, Q 2 Rnet/2 . Hence, we may expect that when in a given vein R > Rnet , it is energetically more favorable to flow fluid through the rest of the network and hence to shrink the vein. Reciprocally, if R < Rnet , we expect that the vein is stable. Analyzing our equations gives further support to this intuitive rule. When R ≫ Rnet , from Equation 3, we may expect ⟨τ ⟩ to be relatively small, in particular, small relative to the vein's specific steady state τ 0 and hence via Equation 1 the vein would likely shrink. Reciprocally, if R ≪ Rnet , we may expect ⟨τ ⟩ to be relatively large compared to its specific τ 0 , and hence the vein is stable. Yet, since R/Rnet is nondimensional, it can provide more systematic insight than ⟨τ ⟩ , since τ 0 is not known a priori. Notice that the red arrow in Figure 4B presents a shrinking vein that indeed verifies R > Rnet . However, according to shear rate measures (see yellow arrow in Figure 1B. ii), the shear rate is large in that vein, preconditioning the vein to grow, according to previous works (Taber, 1998a;Hacking et al., 1996;Hu et al., 2012;Secomb et al., 2013;Pries et al., 1998;Pries et al., 2005;Hu and Cai, 2013). We can therefore show why occasionally, veins at high shear rate shrink, and veins at low shear rate grow by highlighting that R/Rnet , beyond shear rate, is crucial to predict vein fate.
Our aim is now to investigate, in more detail, how these novel feedback parameters integrating network architecture, the relative resistance R/Rnet and the local pressure P via the target shear rate, control vein dynamics on the basis of three key network topologies of a vein.
Specific vein fates: Dangling ends, parallel veins, and loops Dangling ends are unstable: Disappearing or growing As observed in our data, dangling ends are typical examples of veins that can start with very similar shear rate and radius and yet suffer radically different fates ( Figure 1B.i, ii, Figure 5A). Dangling ends either vanish or grow but never show stably oscillating trajectories.
Topologically, and unlike the middle vein considered in Figure 3A, dangling ends are only connected to the rest of the network by a single node. Therefore, the relative resistance Rnet cannot be calculated in a dangling end and cannot play a role. The shear rate in a dangling end is simply ⟨a⟩T . Using this expression instead of Equation 3 and analyzing the dynamical system with Equations 1; 2, we find that dangling veins can only shrink or grow (see Appendix 4). Furthermore, τ 0 determines the threshold for growth over shrinkage. Since τ 0 ∼ τ active − ⟨P − P 0 ⟩/µ a large ⟨P⟩ decreases τ 0 . Hence, the model predicts that a larger pressure at a dangling end facilitates growth.
We observe for the example of Figure 5A that large values of ⟨P⟩ indeed appear to favor growth, and small values prompt veins to vanish. This agrees with physical intuition: when a vein is connected to a large input pressure, one expects the vein to open up. Notice, however, that here the mechanism is subtle. The shear rate itself is not large. Rather, the shear rate threshold to grow is lowered by the high local pressure. Local pressure is thus connected to dangling end fate: it is a prime example of the importance of integrating network architecture.
Competition between parallel veins decided by relative resistance Parallel veins are another example in which initially very similar and spatially close veins may suffer opposite fates; see Figure 5B. Often, both parallel veins will eventually vanish, yet what determines which vanishes first?
To investigate this situation we can simply extend the circuit model of Figure 3A with another parallel resistance, corresponding to the parallel vein (Appendix 4). We then have two veins with respective resistance say R 1 and R 2 . We can analyze the stability of this circuit with similar tools as above. We find that if one vein's relative resistance is larger than the other one's, say for example R 1 /R net,1 > R 2 /R net,2 , then vein 1 vanishes in favor of the other vein 2 as previously predicted in simpler scenarios for steady states (Hacking et al., 1996). Exploring R/Rnet in our full network ( Figure 5B), we find that a vein with a large relative resistance R/Rnet > 1 will vanish. In contrast, a nearby, nearly parallel vein with R/Rnet ≃ 1 will remain stable.
The relative resistance R/Rnet is thus a robust predictor for locally competing veins. Although it is connected to shear rate, as highlighted through Equation 3, there are clear advantages to the investigation of R/Rnet over the shear rate itself: R/Rnet is straightforward to compute from global network architecture as it does not require to resolve flows, and it is non-dimensional.

Loops shrink first in the middle
Finally, loopy structures i.e. a long vein connected at both ends to the remaining network, are often observed in P. polycephalum. Surprisingly, we experimentally observe loops to start shrinking in their very middle ( Figure 5C, Appendix 1- figure 4F and Appendix 1-figure 5F) despite the almost homogeneous vein diameter and shear rate along the entire loop. This is all the more surprising as quantities such as ⟨P⟩ and R/Rnet are also similar along the loop.
This phenomenon again resides in the network architecture, and we can rationalize it with an equivalent flow circuit (see Appendix 4). When a vein segment in the loop shrinks, mass has to be redistributed to the rest of the network. This increases shear rate in the outer segments, preventing the disappearance of the outer segments of the loop. Once the center segment has disappeared, both outer segments follow the dynamics of dangling ends. Their fate is again determined by network architecture, through the local pressure ⟨P⟩ in particular.
Importantly, we find that as soon as a vein disappears, the network's architecture changes: flows must redistribute, and vein connections are updated. Hence, an initially stable vein may become unstable. Vein fates, thus, dramatically evolve over time.

Single vanishing vein triggers an avalanche of vanishing events among neighboring veins
After focusing on individual vein dynamics, we now address global network reorganization. Observing a disappearing network region over time, we find that veins vanish sequentially in time ( Figure 6A, B). Inspired by the importance of relative resistance for parallel veins, we here map out relative resistance R/Rnet at subsequent time points in an entire region ( Figure 6A). At the initial stage ( Figure 6A, 2 min), the majority of veins are predicted to be stable with a relative resistance R/Rnet < 1 . As expected, the few veins with high relative resistance (red arrows in Figure 6A, 2 min) indeed vanish first (black crosses in Figure 6A, 5 min).
As a consequence of veins vanishing, the local architecture is altered, and the relative resistance, through Rnet , changes drastically. Veins that were stable before are now predicted to be unstable. This avalanche-like pattern, in which individual vanishing veins cause neighboring veins to become unstable, repeats itself until the entire region disappears in less than 15 min (Appendix 1- figure 4F and Appendix 1- figure 5F show similar avalanches in other specimens). Note that a vanishing vein may rarely also stabilize a previously unstable vein ( Figure 6A, 16 min, blue arrow).
The fundamental origin of these avalanches of vanishing veins can be narrowed down again to network architecture. We explore a model network region, inspired by a region in an actual specimen ( Figure 6C). We simplify the investigation by considering the region is made of a few veins of similar resistance r connected to the rest of a network, represented by an overall equivalent resistance Rrest . Rrest represents the rest of the network relative to the region, distinct from Rnet , which is relative to a single vein. We precondition all veins to be stable, assuming that for each vein its relative resistance R/Rnet ≲ 1 . Since in our model network for each vein, we approximately have R/Rnet ∼ r/Rrest this prescribes the initial values of r/Rrest ≲ 1 .
We now perturb a vein slightly, for example with a smaller radius, and therefore with a slightly higher resistance, say 2r (purple in Figure 6C). The perturbed vein's relative resistance thus may become greater than 1, making the vein unstable. As the vein vanishes, two network nodes are removed, and individual veins previously connected through the node now become a single longer vein. A longer vein has a higher hydraulic resistance. Hence, the 'new' longer vein also becomes unstable (blue in Figure 6C). Once it vanishes, in turn, another neighboring vein becomes longer and unstable (green in Figure 6C). Reciprocally, vein growth and parallel vein disappearance can -more rarely -decrease R/Rnet , and in turn, stabilize a growing vein, as highlighted by the blue arrow in Figure 6A at 16 min.
In our simple mechanistic model, the series of events follows an avalanche principle similar to that observed in our experiments: a vanishing vein disturbs local architecture. This modifies the relative resistance of nearby veins and hence their stability. The avalanche of disappearing veins eventually results in the removal of entire network regions.

Discussion
We here report highly resolved data of spontaneous network reorganization in P. polycephalum in which both individual vein dynamics and fluid flows pervading veins are quantified simultaneously. We observe disparate vein dynamics originating from shear-driven feedback on vein size. Strikingly, sheardriven feedback occurs with a time delay ranging from 1 min to 3 min. Our vein network model challenges previous concepts showing that vein fate is not only determined through shear rate magnitude but also through parameters that integrate network architecture via fluid flow. In particular, dangling end fate is connected to local fluid pressure ⟨P⟩ , with larger pressures stabilizing dangling ends. Inner network vein fate is tightly determined by the vein's resistance relative to the resistance to fluid flow  through the rest of the network, R/Rnet . When R/Rnet > 1 (reciprocally R/Rnet < 1 ), this preconditions the vein to shrink (respectively to grow or be stable). While R/Rnet is directly related to shear, it can be easily computed from network morphology, without needing to resolve flows. Both relative resistance R/Rnet and local pressure ⟨P⟩ are based on fluid flow physics and are indirect measures of the entire network architecture. Yet, network architecture strongly depends on time. As unstable veins vanish, the relative architecture of changes, inducing avalanches of vanishing veins, resulting in significant spontaneous reorganization.
While our experimental investigation is specific to P. polycephalum, we expect that the two key concepts unraveled here, time delay and network architecture governing vein fate through relative resistance and fluid pressure, may very well be at play in other vascular networks. First, the ubiquity of delayed shear rate feedback, beyond the contractile response of actomyosin, suggests that a diversity of vein dynamics (circling, non-monotonic) may also occur in other vascular networks. In fact, also the turnover time for actin filaments in living cells ranges from 10 s to 30 s, close to our measured time delay (Fritzsche et al., 2013;Livne et al., 2014;Colombelli et al., 2009). Other pathways, such as chemical pathways for sheared endothelial cells in blood vasculature, are processed with a time delay of a few minutes (Lu and Kassab, 2011;Godbole et al., 2009;Fernandes et al., 2018), while reorganization occurs on longer timescales ranging from 15 min for individual cells to several days for blood vasculature (Livne et al., 2014;Landau et al., 2018).
Second, network architecture feedback, through relative resistance and pressure, is connected to the laminar flows pervading the network. Thus, our perspective could be extended to other networks where laminar flows are an essential building block, in essence, to the diversity of networks where Murray's law holds at steady state (West et al., 1997;Kassab, 2006;McCulloh et al., 2003;Akita et al., 2017;Fricker et al., 2017). Particularly, our insight suggests simple parameters to map out, such as the purely geometrical relative resistance. Likely these parameters, which integrate network architecture, may explain discrepancies between shear rate and network reorganization in other vascular networks (Chen et al., 2012;Baumgarten and Hauser, 2013;Rosenfeld et al., 2016;Chang and Roper, 2019;Sugden et al., 2017).
Notably, imaging of biological flow network as a whole is, as of now, a rare feature of our experimental system that enabled us to unravel the importance of the network architecture for vein fate. Yet, we are hopeful that future theoretical work may allow for vein fate prediction with relative resistances determined only with partial information of a network's architecture, with sufficient accuracy. At the same time, novel experimental techniques now open up the way for in toto imaging of vascular systems and quantitative assessment of dynamics (Daetwyler et al., 2019).
The fact that pervading flows and network architecture are so intermingled originates in the simple physical principle that flows are governed by Kirchhoff's laws at nodes, and hence 'autonomously' sense the entirety of the network's architecture. Yet, Kirchhoff's laws are not limited to flow networks, but also govern electrical (Dillavou et al., 2022), mechanical (Hexner et al., 2018;Goodrich et al., 2015;Berthier et al., 2019b;Berthier et al., 2019a), thermal (Chen et al., 2015 and resistor-based neural networks (Erokhin et al., 2010;Li et al., 2018). Having the physics of Kirchhoff-driven selforganization at hand may thus pave the way for autonomous artificial designs with specific material (Hexner et al., 2018;Goodrich et al., 2015) or learning properties (Dillavou et al., 2022;Erokhin et al., 2010;Li et al., 2018).
Rrest represents the rest of the network relative to the region.In this model, a vein vanishes if its individual relative resistance R/Rnet > 1 . The disappearance of veins sequentially increases the relative resistance of neighboring veins, making them unstable. Here r/Rrest = 0.1 , yet similar behavior was obtained consistently over a wide range of r values. Preparation and imaging of P. polycephalum P. polycephalum (Carolina Biological Supplies) networks were prepared from microplasmodia cultured in liquid suspension in culture medium (Li et al., 2018;Fessel et al., 2012). For the full network experimental setup, as in Figure 1B of the main text (see also Video 2, Appendix 1-Video 7, and Appendix 1-Video 8) microplasmodia were pipetted onto a 1.5% (w/v) nutrient free agar plate. A network developed overnight in the absence of light. The fully grown network was trimmed in order to obtain a well-quantifiable network. The entire network was observed after 1 h with a Zeiss Axio Zoom V.16 microscope and a 1 x/0.25 objective, connected to a Hamamatsu ORCA-Flash 4.0 camera. The organism was imaged for about an hour with a frame rate of 10 fpm.

Additional information
In the close-up setup, as in Figure 1A of the main text (see also Video 1, Appendix 1-Video 1, Appendix 1-Video 2, Appendix 1-Video 3, Appendix 1-Video 4, Appendix 1-Video 5 and Appendix 1-Video 6) the microplasmodia were placed onto a 1.5% agar plate and covered with an additional 1 mm thick layer of agar. Consequently, the network developed between the two agar layers to a macroscopic network which was then imaged using the same microscope setup as before with a 2.3 x/0.57 objective and higher magnification. The high magnification allowed us to observe the flow inside the veins for about one hour. Typical flow velocities range up to 1 mms -1 (Bykov et al., 2009). The flow velocity changes on much longer timescales of 50 s to 60 s. To resolve flow velocity over time efficiently 5 frames at a high rate (typically 60 ms, detailed frame rates are specified for each Video) were imaged separated by a long exposure frame of about 2 s. As different objectives were required for the two setups, they could not be combined for simultaneous observation. Typically the longer exposure frame appears as a bright flash in the Videos. The 12 close-up data sets are indexed #A-L consistently in the main text and Appendix.

Image analysis
For both experimental setups, image analysis was performed using a custom-developed MATLAB (The MathWorks) code. This procedure extracts the entire network information of the observed organism (Bäuerle et al., 2017): single images were binarized to identify the network's structure, 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. https:// elifesciences. org/ articles/ 78100/ figures# video1 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. https:// elifesciences. org/ articles/ 78100/ figures# video2 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.

https:// elifesciences. org/ articles/ 78100/ figures# video4
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. https:// elifesciences. org/ articles/ 78100/ figures# video5 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. https:// elifesciences. org/ articles/ 78100/ figures# video6 using pixel intensity as well as pixel variance information, extracted from an interval of images around the processed image. As the cytoplasm inside the organism moves over time, the variance gives accurate information on which parts of the image belong to the living organism and which parts are biological remnants. The two features were combined and binarized using a threshold. The binarized images were skeletonized and the vein radius and the corresponding intensity of transmitted light were measured along the skeleton. The two quantities were correlated according to Beer-Lambert's law and the intensity values were further used as a measure for vein radius, as intensity provides higher resolution. For the imaging with high magnification, in addition to the network information, the flow field was measured using a particle image velocimetry (PIV) algorithm inspired by Thielicke and Stamhuis, 2014b;Thielicke and Buma, 2014a;Thielicke and Stamhuis, 2014c, see Figure 1A. ii of the main paper. The particles necessary for the velocity measurements are naturally contained within the cytoplasm of P. polycephalum.

Flow calculation from vein contractions
Building on the previous image analysis, we used a custom-developed MATLAB (The MathWorks) code to calculate flows within veins for the full networks, based on conservation of mass. The algorithm follows a two stage process.
First, the network structure obtained from the images was analyzed to construct a dynamic network structure. This structure consists in discrete segments that are connected to each other at node points. At every time point, the structure can evolve according to the detected vein radii: if a radius is lower than a certain threshold value, the corresponding segment vanishes from the structure. Segments which are isolated due to vanishing segments are also removed. We carefully checked by eye that the threshold levels determining when a segment vanished agreed with brightfield observations. Note that we do not account for entirely new segments in the dynamic structure. As no substantial growth occurs in our data, this is a good approximation.
Second, flows and pressure in each segment were calculated building on Alim et al., 2013. We formalize this step briefly. Let n and p be two indices to describe node n and node p connected by a segment say i . In each segment, there is an unknown inflow from neighboring segments Q 0,np . There is also added flow arising due to periodic contractions Q in,np = 2πa i L i ∂ai ∂t where a i denotes the radius of segment i and L i is the length of the vein. Note that all flows are given directed from node n to node p . As a result the flow arriving from segment i at node p is simply Q 0,np + Q in,np . According to Kirchhoff laws, at each node in the network, at each time point, the total incoming flux from each segment has to be zero This can be rewritten where Qp are the new unknowns. Since Poiseuille law holds, the Q 0,np are given by where Pn is the local pressure at node n (respectively Pp at node p ) and Rnp is the hydraulic resistance of a vein such that Rnp = πa 4 i /8µL i . Hence which is a linear equation of the form Q =ḠP where P is the vector of pressure at each node in the network, similarly Q is the vector of unknown inflows at each node, and Ḡ is a matrix of inverse resistances taking into account the architecture of the network. We can invert this equation to obtain the values of pressure at network nodes. Then we calculate the inflow from neighboring veins through Equation A1.3. Finally, we obtain pressure in segment i as P i = (Pn + Pp)/2 . Compared to Alim et al., 2013, we introduced two major additions. On the one hand, the actual live contractions a i (t) are used, as detected from sequential images. To ensure that Kirchhoff's laws are solved with a good numerical accuracy, the radius traces a i (t) were (1) adjusted at each time so that overall cytoplasmic mass is conserved (mass calculated from image analysis varied by less than 10% over the analysis time) and (2) overdiscretized in time by adding 2 linearly interpolated values between each frame. Hence the simulation time step ∆t = 2 s is 3 times smaller than the acquisition time, and favors numerical convergence of all time dependent processes. Note that the results were found to be independent of the simulation time step ∆t when decreasing it by a factor 2. On the other hand, a segment (or several) that vanishes creates (just before disappearing) an added inflow of −πa 2 i L i /∆t , where a i the segment's radius just before disappearing. This corresponds to radius retraction as observed in the data.

Data analysis -time averages
For all data, we extract short time averages by using a custom-developed MATLAB (The MathWorks) routine. To determine the short time averages of the oscillating shear rate and vein radius, we used a moving average with a window size of tave ≃ 2 − 3T ( T ≃120 s ). The i th element of the smoothed signal is given by At the boundary where the averaging window and the signal do not overlap completely, a reflected signal was used as compensation. This can be done because the averaging window is relatively small and the average varies slowly in time. The determined trend (for the close-up data sets) was then smoothed with a Gaussian kernel to reduce artefacts of the moving average filter.
In experimental data of the shear rate, we observe that raw shear rate appear to oscillate at rather high frequency (see e.g. Figure 1iii). Here we briefly rationalize this behavior. First a zoom in time of the data in Figure 1A. iii, see Appendix 1-figure 1, shows that in fact the frequency at which raw shear rate oscillates is double that of the frequency of oscillations of the vein radius. We explain this frequency doubling based on a minimal example. Consider a minimal example with a contraction pattern a(t) ≃ ⟨a⟩(t)(1 + ϵ cos(2πt/T)) , where the average radius slowly evolves in time as ⟨a⟩ = L cos(2πt/t adapt ) . The flow in the vein is Q = L d(πa 2 ) dt and therefore the shear rate at lowest order in ϵ is The resulting shear rate contains the absolute value of a periodic quantity of period T , hence, is periodic with half the period T/2 . We plot the minimal example curves in Appendix 1-figure 1B. We further check whether our algorithm to extract the shear rate trend is correct even on these high frequency raw data. Averaging the raw shear rate obtained in the above minimal model over one contraction period yields which is exactly the amplitude of the raw τ data up to a constant numerical prefactor. Hence, our averaging is well suited to extract reliable trends of the shear rate. In Appendix 1-figure 1B, we present the results from our averaging algorithm (full thick red line) and the theoretically calculated trend (yellow dashed) and obtain excellent agreement. Our time-averaging algorithm is therefore well-suited to the investigation of even these high frequency data.

Data analysis -Additional shear rate -radius data
To add to the data presented in Figure 1iv presenting the time-averaged dynamics of radius adaptation and shear rate, we show in Appendix 1-figure 2 (resp. Appendix 1- figure 3) additional dynamics for the close-up datasets (respectively the full network #1 of Figure 1B).

Data analysis -Additional data on full networks Additional data on different full network specimen
In what follows we present additional data on full networks. In particular, we investigate two other full networks besides specimen #1 (of Figure 1B), which we call #2 and #3. These two additional networks show significant spontaneous reorganization over time and we show snapshots of their initial and final networks in Appendix 1- figure 4A, B and Appendix 1- figure 5A, B.
We also present additional data to demonstrate the existence of similar ambiguity in shear rate -radius response in other full networks. We show with yellow arrows additional places where shear rate is initially high yet the vein will disappear in Appendix 1- figure 4C and Appendix 1-figure 5C. Red dots in Appendix 1- figure 4B and Appendix 1-figure 5B also show veins where shear rate is initially low however these veins will grow in time.
We present pressure data in Appendix 1- figure 4D and Appendix 1- figure 5D. We find that pressure doesn't vary much throughout the network. A global pressure wave is observed corresponding to a stable direction of the peristaltic contractions. We identify in these maps nearby veins and find that the ones with larger pressure remain (blue stable) while those with lower pressure vanish (pink unstable).
We present relative resistance data R/Rnet in Appendix 1- figure 4E and Appendix 1- figure  5E. We find a number of veins with R/Rnet > 1 , indicated by pink arrows, that indeed vanish in time.
To finish with the analysis of additional networks, we present a map of the time of disappearance of veins in the full specimen in Appendix 1- figure 4F and Appendix 1-figure 5F. We find that loops consistently vanish by their center, as highlighted via black arrows.

Additional data on full network specimen #1
In Appendix 1- figure 6 we present additional data on Specimen #1 that is the main example under scrutiny in the main text. We provide in particular maps of quantities that are not shown in the main text, such as the connected resistance Rnet (C) and Q in =  We also provide cross-correlation data between specific quantities and initial vein radius ⟨a⟩ at the beginning of the experiment, Rnet , R/Rnet , Q in and ⟨P⟩ (A,B,D,E). We find that the only quantity that is significantly correlated with ⟨a⟩ is Q in , coherently since we expect Q in ∝ ⟨a 2 ⟩ .
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. 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 have short timescale variations when they have significant variations over timescales much smaller than the peristaltic contractions T ≃ 1 − 2 min ; and long timescale variations if they vary over longer timescales corresponding to vascular adaptation and rearrangement. We briefly summarize here the derivation of our vascular adaptation model from force balance and provide more details in our accompanying publication (Marbach et al., 2023). We consider the force balance equation on a small vein wall segment of radius a , length L , thickness e . As the wall motion is typically slow and occurring over microscopic scales we neglect inertial contributions and write where P − P 0 is the hydrodynamic pressure difference between interior and exterior, σ circum is the circumferential stress (or elastic tension), σ active corresponds to active stresses from the actomyosin cortex (Radszuweit et al., 2013;Alonso et al., 2017), and γL da dt is the friction force reflecting the long timescale for fiber rearrangement (Salbreux et al., 2012;Fischer-Friedrich et al., 2016). Note that since the shear rate τ acts longitudinally on the walls, it does not contribute to the force balance on the radial direction. Yet, the vein walls consist of a material with an anisotropic response to shear, namely cross-linked fibers (the actomyosin gel). Hence, when sheared, a radial stress σr(µτs) builds up as a result of longitudinal shear rate sensing (with a time delay) (Gardel et al., 2008;Janmey et al., 2007;Vahabi et al., 2018;Lu and Kassab, 2011;Godbole et al., 2009;Fernandes et al., 2018).
The general force balance (2.1) significantly simplifies when we average over the short timescales of vein contractions (1-2 min) (Isenberg and Wohlfarth-Bottermann, 1976), typically corresponding to elastic deformations, to focus on the longer timescales of 10-60 min corresponding to vein wall assembly or disassembly inherited from e.g. actin fiber rearrangements (Salbreux et al., 2012;Fischer-Friedrich et al., 2016).
On these longer timescales, significant morphological vein adaptation of ⟨a⟩ occurs. ⟨σ active ⟩ is a constant as it is expected to vary only on short timescales in line with the periodic contractions. Note also that it is a negative stress, that tends to shrink a vein -this reflects the impact of metabolic cost, here induced by vein wall activity. ⟨σ circum ⟩ ≃ 0 over short timescales, as such forces are intrinsically elastic forces and hence do not pertain long time features. Finally, our numerical calculations of pressures within observed networks show that ⟨P − P 0 ⟩ depends smoothly on the location within the network, but barely varies in time (Alim, 2018; Figure 4A). We obtain a time-independent, yet position-specific constant τtarget = − 1 µ ⟨P − P 0 ⟩ + τ active , where we wrote τ active = ⟨σ active ⟩/µ . Furthermore, we assume a phenomenological functional form for the radial stresses, as σr(µτs) ≃ µ τ 2 s τc , in line with observations of sheared cross-linked actin fibers (Gardel et al., 2008;Janmey et al., 2007) where τc is a positive constant. Importantly, this radial stress, acts in the positive direction, i.e. dilates vessels. This functional form is also consistent with measured data on fibrin gels (Vahabi et al., 2018;Kang et al., 2009) and models of anisotropic response based on nonlinear elastic theory (Vahabi et al., 2018).
Finally, to simplify the expressions we now introduce τ 0 = √ τcτtarget and a characteristic adaptation timescale for vascular rearrangement. This allows us to recover the vascular adaptation rule Equation 1. While the two parameters τ 0 and t adapt may appear to be coupled at the scale of the network, there is actually no reason for τc or for γ to be constant throughout the network. In fact they may very well depend on the age of the vein, the absolute thickness of the actomyosin gel, etc. Again, we refer the reader to more details on the derivation in our accompanying manuscript (Marbach et al., 2023). Agreement with Murray's law Our model is consistent with Murray's steady state assumption. In fact, the (non-trivial) steady state of our model Equations 1; 2 corresponds to a constant average shear in the vein ⟨τ ⟩ = τ 0 . This corresponds exactly to Murray's result of minimum work.
In fact, Murray stipulates that the energy dissipation E of a single vein (of radius a and length L ) is given by flow dissipation associated with the vein's resistance and energy expense to sustain the vein where R = 8µL/πa 4 is the vein resistance assuming Poiseuille flow in the vein, b is a local metabolic constant per unit volume, Q the flow rate and µ viscosity. The principle of minimum energy expense suggests to search for the minimum of E with respect to the vein radius a which gives the relation a 6 optimal = 8Q 2 η bπ 2 . The shear rate τ can be expressed as τ = 4Q πa 3 and hence the optimal (or steady state) shear rate is independent of radius and flow rate τ optimal = √ b/µ . This is consistent with our steady state where shear rate is constant ⟨τ ⟩ = τ 0 . The constant τ 0 can thus also be interpreted as being related to the typical local energy expense to sustain the vein √ b/µ (which corresponds very closely to our τ active characterizing metabolic expense to sustain the contractile activity). Note that we bring further insight compared with Murray's derivation, as our adaptation dynamics (2.1) originates from force balance on the vein wall, and hints that τ 0 (or the metabolic cost) also depends on local pressure ⟨P⟩ .

Extracting the time delay from data analysis
In this section we discuss our procedure to extract the time delay from data.
First, we verify that the time delay we extract is independent of the averaging technique. To do so, we investigate the time delays obtained from the cross-correlation of da/dt and τ instead of their averaged counterparts d⟨a⟩/dt and ⟨τ ⟩ . We obtain a distribution of best time delays, over the nearly 10000 vein segments of the full network, and we retain maxima regardless of the value of the crosscorrelation. We present the results in Appendix 2- figure 1A. The average time delay is 52 s , which is comparable in orders of magnitude to the average time delay of 122 s for the same network data but where radii and shear rate trends were extracted ( Figure 2C). Note, that the correlation however is much less clear without extracting trends and in average the correlation score is 0.25 with only 5% of veins achieving a score gt 0.2 compared to 0.66 average score with trends with 15% of veins achieving a score gt 0.5 . Note that the average correlation is quite low because in general the data are 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 trends, (ii) extracted trends and best time delay obtained shown on trends and (iii) cross correlation between da/dt(t) and τ (t − t delay ) with respect to the searched time delay t delay , and maximum value shown. not perfectly periodic and smooth. Hence, we decide to keep the analysis on the data trends, that appears to be much more precise.
Second, we check that even if the time delay between adaptation and shear rate is close to the peristaltic contraction frequency ( T ≃ 1 − 2 min ), we are still able to extract it with our method reliably. To do so, we consider model data a(t)/a 0 = (1 + 0.2 sin[ωs(t − t delay )])(1 + 0.2 cos[ω(t − t delay )]) and τ (t)/τ 0 = (1 + 0.2 cos[ωs(t)]) cos[ω(t)] . We impose a contraction period T = 2π/ω = 120 s and the long time adaptation period 2π/ωs = 20 min , and a delay similar to the beating period t delay = T = 120 s . Using our methodology to extract the time delay, we find t delay = 114 s , which is equal to the set time delay of 120 s within the error bar of 6 s corresponding to the time step where data was sampled. We conclude that the time delay we obtain is independent of the value of the contraction frequency.
Finally, some trajectories appear to oscillate on long timescales say with period Tosc . Hence, it may not be obvious by cross-correlation for these specific trajectories to determine whether the delay is t delay or −Tosc + t delay , or another combination. Tosc characterizes rarely observed long cycles in the long time adaptation dynamics, for example see Figure 2D, and typically Tosc = 20 min . In contrast, the apparent phase lag between ⟨τ ⟩ and ⟨a⟩ is usually of the order of a few minutes in the samples where the delay can be inferred unambiguously ( t delay ∼ 1 − 5min ). We may thus expect that the time delay is indeed t delay ∼ 1 − 5min and not −Tosc + t delay which would be much longer. We impose this condition by adding a cutoff on the time delay at 5 min. Changes to the time delay cutoff, for example setting the cutoff to 10 min, does not affect the results significantly. In fact, strictly oscillatory signals are very rare. For example Figure 1B. iii clearly shows a lag time (between 7-15 min) that allows one to resolve the causality relation unambiguously.

Time delays in close-up and full networks -additional data
We now present time delay analysis in all our specimens.
In Appendix 2-figure 2 we present time delay data in full networks. Time delays (both positive and negative) were retained for veins for which the maximum cross-correlation was higher than 0.5. Time delays may only be extracted with sufficient accuracy for stable veins, which do not represent the majority of veins in the network. Hence approximately 15 -25% of observed veins reach a significant cross-correlation and allow us to record a value of the time delay. To avoid biasing the statistical search with either positive or negative time delays, we allow the algorithm to record both positive and negative time delays for a single vein if these maxima are significant. The phenomenon of negative time delays is quite infrequent. For example in specimen #1, out of the observed veins that yield a time delay, we find 94% with a positive time delay, 4% with a negative time delay, and 2% with both a positive and negative time delay. For specimen #2 we find 96% positive, 3% negative and 1% positive and negative, and for specimen #3 respectively 87%, 12% and 1%. Hence, positive time delays are much more likely. The average time delay is consistently t delay ≃ 2 min .
We also investigate the time delay on close-up data sets (see Appendix 2- figure 3), and only on stable close-up data sets as they will allow us to extract the time delay more reliably. Notice that cross correlations are usually quite smooth as the correlation continuously increases until significant shear rate and radius changes are aligned. The correlation maximum corresponds to a strongly correlated configuration (gt 0.7 ). Vein #E finds a best time delay that is quite large ( t delay ∼ 12 min ), potentially due to the fact that we are exploring a very long time sequence for this particular vein and that the cross correlation algorithm picks up a large change unrelated to the actual short delay. Notice, however, that a time delay of 2-5 min potentially corresponding to the cross-correlation shoulder also seems suited here. The variability in the time delay extracted on close-up data sets show the need for statistical analysis of the time delay, which we perform on full networks.

Fitting of the model to data
Fitting of the model Equations 1; 2 to the data was performed using a non-linear least squares algorithm included in the SciPy optimize package (Virtanen et al., 2020), or a linear least squares algorithm, according to whether two or three model parameters had to be fitted. The relative fitting error is defined as where Nt is the number of data points.
First, we fit close-up data sets, for all three parameters t adapt , t delay and τ 0 . As stressed in the main text the model parameters are not expected to be constant over long times (on which loopy trajectories are typically observable). To find suitable time frames where model parameters where approximately constant and loopy trajectories observable, we systematically varied the time windows of the data used for fitting. To find the optimal time windows for fitting including fitting the time delay t delay , we chose close-up data sets forming loopy trajectories (#G, #E, #F and #K), as loops are a characteristic feature ensuing from the time delayed dynamics. The distribution of time delays fitted for different time windows was found to range from 1 min to 10 min (see Appendix 2- figure 4), which is within the range obtained via the cross-correlation algorithm described above in Appendix 2.3. Fitted trajectories reproduce the main features observed experimentally in detail. The corresponding fitted parameters are reported in Appendix 2-table 1.
Second, we fit all close-up data sets now only including two model parameters, τ 0 and t adapt . We fixed the time delay to a constant value t delay =120 s . We fit different time intervals in the data sets and find very good agreement between data and fits -see Appendix 2-figure 5. We report the corresponding fitted parameters in Appendix 2-table 2.
Finally, we fit a random sample of 15 veins from the full network specimen #1. We include two model parameters, τ 0 and t adapt . We fixed the time delay to the value obtained by cross correlation. We fit only over one rather larger time interval of about 40 min and find reasonable agreement between data and fit (see Appendix 2- figure 6). The corresponding fitted parameters are reported in Appendix 2-table 3. In addition for the 15 veins from the full network we also fit the model with no time delay t delay = 0 . For these fits, we set τs = ⟨τ ⟩ instead of Equation 2. We show the fitted results in black dotted lines in Appendix 2- figure 6 and report here only the corresponding fitting error ϵerr in Appendix 2-table 3. We find a systematic higher fitting error for fits without time delay over those with time delay.  of the equivalent resistance however the vein in series does. We have one vein in series of resistance R with two veins in parallel of resistance R . The equivalent resistance is Rnet = R + R/2 = 3R/2 > R and the vein is a priori stable now. Since this network is slightly more complex we can investigate the fate of other veins in that same network, which we do in Appendix 3-figure 1E, F. We find that these other veins are unstable or stable. This shows that even in a simple network, the relative resistance is a key measure to discriminate between different veins.

Generic flow network equivalent circuit
We focus on the generic flow network equivalent circuit as given in Figure 2A. iii of the main paper and derive the circuit laws as given in the text -see also Appendix 4- figure 1A that recapitulates notations.
Because of Kirchhoff's laws we easily find that Q in = −Qnet . Then we look for the value of the flow rate flowing through the vein of interest Q . We see that Analysis of the feedback system between shear rate and vein radius In the main text, we have established a set of coupled equations describing the adaptation of veins in a dynamic network. The specific form of these dynamic equations depends on the position of the considered vein within the network. In this section we discuss the stability of a vein fully connected to the network (generic flow network equivalent circuit). Note that other cases (dangling ends, loops, parallel veins) can be easily discussed with similar methodologies.
To simplify the discussion of the fixed points of the dynamical system (⟨τ ⟩, τs, ⟨a⟩) , it is equivalent to study the fixed points of (τs, ⟨a⟩) , taking into account Equation 3 in Equation 2. The dynamic system of equations is then given by: since Qnet = 8πLϵ⟨a⟩ 2 /T where ϵ is the characteristic contraction percentage of the vein (dimensionless). Plotting the nullclines of Equations A3.5 and A3.6 in the (τs, ⟨a⟩) space, we observe one, two or no intersections of the nullclines, which correspond to fixed points of the system, depending on the physical parameters. In particular, there is a fixed point corresponding to a vanishing vein in . In the following, we will investigate the conditions for the existence and the stability of these fixed points.
Linear stability of the feedback-system The dynamical system defined in Equations A3.5 and A3.6 has up to three fixed points. To analyze the stability of those fixed points we use linear stability analysis (Strogatz, 1994;Argyris et al., 2017). The first fixed point is at (τs = 0, ⟨a⟩ = 0) , the other two are defined by (τs = τ 0 , ⟨a⟩ = r 0,± ) , where r 0,± are the real positive solutions of the equation τ 0 = ⟨τ ⟩(r 0 ) . To analyze the stability of those fixed points, we calculate the Jacobi matrices J at each location: For (0, 0) the eigenvalues can be read off from the Jacobi matrix as λ 0,1 = − 1 tadapt and λ 0,2 = − 1 tdelay . Consequently, the fixed point is stable, as all model parameters are positive. The two other fixed points, as mentioned above depend on the root ⟨a⟩ 0 of τ 0 = ⟨τ ⟩(⟨a⟩ 0 ) .
The stability of those fixed points is therefore conditional on the value of these roots. To gain insight on the stability of the fixed points we look at the two extreme cases of either small or large tube radii (as specified below). We will then extend our insight to intermediate tube radii.
• ⟨a⟩ → ∞ : In the case of a large tube radius, the shear rate simplifies to ⟨τ ⟩(⟨a⟩) = 32Lϵ T 1 ⟨a⟩ and we find only one fixed point at (τs = τ 0 , ⟨a⟩ = 32Lϵ Tτ0 ) . The Jacobian at this fixed point is with the eigenvalues λ 2,± = − √ t adapt ± √ t adapt − 8t delay 2 √ t adapt t delay (A3.14) We now have to differentiate two cases. The first one is t adapt > 8t delay . Then λ 2,± < 0 and the fixed point is stable. For the case t adapt < 8t delay , we have ℜ(λ 2,± ) < 0 , ℑ(λ 2,± ) ̸ = 0 and the fixed point is a stable spiral, which introduces an additional rotation to the system's trajectories. To investigate the direction of the rotation of this hypothetical spiral, one can look at the sign of Equation A3.5 for positive displacements δ along the shear rate axis. We find that d⟨a⟩ dt (τ s =τ 0 +δ,⟨a⟩=⟨a⟩ 1 ) = ⟨a⟩ 1 t adapt τ 0 δ + δ 2 τ 2 0 > 0 (A3.15) and therefore the spiral rotates in the clockwise direction. In summary, our stability analysis has shown that the system has up to three fixed points. Two of them are stable and separated by a saddle point. The qualitative stability in the limiting case is also valid for intermediate tube radii, as the stability of a fixed point only changes when two fixed points collide, which is only the case at the bifurcation point (when ⟨a⟩ 0 corresponds to the maximum of τ (⟨a⟩) ) (Strogatz, 1994).