We observe vein dynamics in P. polycephalum specimen using two complementary imaging techniques, either close-up observation of single veins or full network imaging (Figure 1 and additional methods in Appendix 1). Close-up vein microscopy over long timescales (Figure 1A.i, see also Video 1 and Video 2) 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 $Q(t)=2\pi \int v(r,t)rdr$. 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.

Figure 1

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Video 1

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Video 2

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Video 3

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Our imaging techniques resolve vein adaptation over a wide range of vein radii, ${\displaystyle a=5-70\mu m}$. Radii data show rhythmic peristaltic contractions, with a period of $T\simeq 1-2\min$ (light blue in Figure 1iii). We calculate shear rate from fluid flows as $\tau =\frac{4}{\pi }\frac{|Q|}{a^3}$. 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 $\tau$ 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 $\tau$, oscillation periods are even doubled due to taking the absolute value of $Q$ in calculating $\tau$; see also Appendix 1—figure 1.

To access the long-time behavior of veins, we average out short timescales on the order of $T\simeq 1-2\min$ 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 $⟨\tau ⟩$ 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).

We relate time-averaged shear rate to time-averaged vein radius and find diverse, complex, yet reproducible trajectories (Figure 1A, B.iv, see also Appendix 1—figure 4 and Appendix 1—figure 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.

We further test this potential causality link between shear rate and vein adaptation. Based on previous theoretical 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; Tero et al., 2007), we expect that the 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), $R=\frac{8\mu L}{\pi {⟨a⟩}^4}$, where ${\displaystyle \mu }$ 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.

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, $\frac{d⟨a⟩}{dt}$, that is the change in time of the vein radius $⟨a⟩$, is related to shear rate $⟨\tau ⟩$ via

Here, ${\tau }_s(⟨\tau ⟩)$ is the shear rate sensed by a vein wall and is directly related to fluid shear rate $⟨\tau ⟩$, in a way that we specify in the following paragraph. The parameter ${\displaystyle t_{\mathrm{a}\mathrm{d}\mathrm{a}\mathrm{p}\mathrm{t}}}$ is the adaptation time to grow or disassemble vein walls corresponding to fiber rearrangement (Salbreux et al., 2012; Fischer-Friedrich et al., 2016) and ${\tau }_0$ the vein’s reference shear rate, corresponding to a steady state regime ${\tau }_s={\tau }_0$ with constant shear rate – in agreement with Murray’s law (see Appendix 2.1; Murray, 1926). ${\displaystyle t_{\mathrm{a}\mathrm{d}\mathrm{a}\mathrm{p}\mathrm{t}}}$ and ${\tau }_0$ 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 $\frac{d⟨a⟩}{dt}$ that we are interested in vascular adaptation, that is on long-time changes in the vein radius. In contrast, the short timescale variations $\frac{d(a-⟨a⟩)}{dt}$ 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, ${\tau }_s(⟨\tau ⟩)$, which itself depends on the average shear rate $⟨\tau ⟩$. 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 $f({\tau }_s)$. We here employ a functional form with a quadratic scaling of the right-hand side on the shear rate $f({\tau }_s)\propto {\tau }_s^2$ 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 $f({\tau }_s)$, as long as $f$ increases with $|{\tau }_s|$ and their exists a non-zero value of shear rate ${\tau }_0$ corresponding to Murray’s steady-state, that is such that $f({\tau }_0)=0$.

Regarding the interpretation of the sensed shear rate ${\tau }_s$, 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, ${\tau }_s$ could correspond to a delayed shear rate compared to the actual one $⟨\tau ⟩$. We turn to confirm this assumption and analyze this time delay further.

We systematically investigate the time delay between shear rate $⟨\tau ⟩$ and vein adaptation ${\displaystyle \frac{d⟨a⟩}{dt}}$. For each vein segment, we calculate the cross-correlation between averaged shear rate $⟨\tau ⟩(t-t_{\mathrm{delay}})$ and vein adaptation $\frac{d⟨a⟩}{dt}(t)$ as a function of the delay ${\displaystyle t_{\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{y}}}$ (Figure 2A). Then, we record the value of ${\displaystyle t_{\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{y}}}$ 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 ${\displaystyle t_{\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{y}}}$ (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 ${\displaystyle t_{\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{y}}}$ is different from ${\displaystyle t_{\mathrm{a}\mathrm{d}\mathrm{a}\mathrm{p}\mathrm{t}}}$. Although both timescales are relevant to describe adaptation in our specimen: ${\displaystyle t_{\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{y}}}$ represents the time to sense shear rate signals in vein walls; ${\displaystyle t_{\mathrm{a}\mathrm{d}\mathrm{a}\mathrm{p}\mathrm{t}}}$ represents the time to grow or disassemble a vein wall.

Figure 2

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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 $t_{\mathrm{delay}}\simeq 2\min$ (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 $t_{\mathrm{delay}}\simeq 2\min$ appears close to the contraction period $T\simeq 1-2\min$. 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.

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 ${\displaystyle t_{\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{y}}}$ explicitly between the shear rate sensed by a vein wall ${\tau }_s$ and fluid shear rate $⟨\tau ⟩$. To this end, we use the phenomenological first-order equation

At steady state, we recover a constant shear rate $⟨\tau ⟩={\tau }_s={\tau }_0$, 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 $⟨\tau ⟩(t)$ as input and fit model constants ${\displaystyle t_{\mathrm{a}\mathrm{d}\mathrm{a}\mathrm{p}\mathrm{t}}}$ and ${\tau }_0$ to reproduce radius data $⟨a⟩(t)$. Note, that ${\displaystyle t_{\mathrm{a}\mathrm{d}\mathrm{a}\mathrm{p}\mathrm{t}}}$ and ${\tau }_0$ 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 ${\displaystyle t_{\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{y}}}$ 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 ${ϵ}_{\mathrm{err}}=\int dt\frac{|⟨a⟩-{⟨a⟩}^{\mathrm{fit}}|}{⟨a⟩}\simeq 0.001-0.17$. 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 ${ϵ}_{\mathrm{err}}$ (see Figure 2D, dotted black line and Appendix 2—table 3).

In all samples, fitting parameters resulted in physically reasonable values. We found $t_{\mathrm{adapt}}\simeq 10-100\min$ corresponding to long timescale adaptation of vein radii. Note again, the physical difference between the time to adapt vein radius ${\displaystyle t_{\mathrm{a}\mathrm{d}\mathrm{a}\mathrm{p}\mathrm{t}}}$ and the time delay to sense shear rate ${\displaystyle t_{\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{a}\mathrm{y}}}$ also translates to orders of magnitude differences with $t_{\mathrm{delay}}\simeq 2\min$ and $t_{\mathrm{adapt}}\simeq 10-100\min$. 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 ${\tau }_0$ it is a priori hard to estimate which values to expect since ${\tau }_0$ is only reached at steady state. Yet, in our continuously evolving specimen, we never reach steady state and, hence, can not measure ${\tau }_0$. However, we can compare ${\tau }_0$ 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 ${\displaystyle t_{\mathrm{a}\mathrm{d}\mathrm{a}\mathrm{p}\mathrm{t}}}$ and ${\displaystyle {\tau }_0}$ 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, ${\displaystyle t_{\mathrm{a}\mathrm{d}\mathrm{a}\mathrm{p}\mathrm{t}}}$ and ${\tau }_0$ are no longer constant for time frames $\gtrsim 40\min$.

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.

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 $⟨\tau ⟩$. Since $⟨\tau ⟩=\frac{4|Q|}{\pi {⟨a⟩}^3}$, 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 $R_{\mathrm{net}}$ parallel to the single vein of $R=\frac{8\mu L}{\pi {⟨a⟩}^4}$ considered within an equivalent flow circuit, see Figure 3A. $R_{\mathrm{net}}$ 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). $R_{\mathrm{net}}$ 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).

Figure 3

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The time-averaged net flow generated by the vein contractions is $Q_{\mathrm{in}}=⟨\left|L\frac{d(\pi a^2)}{dt}\right|⟩\simeq \frac{8\pi Lϵ{⟨a⟩}^2(t)}{T}$ where $ϵ$ is the relative contraction amplitude. The absolute values in this definition are used to measure the net flow. $Q_{\mathrm{in}}$ thus measures the mass exchanges between the network and the vein. As mass is conserved, this results in an inflow of $Q_{\mathrm{net}}=-Q_{\mathrm{in}}$, 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: $QR=-(Q+Q_{\text{net}})R_{\text{net}}$. We, thus, obtain that the time-averaged shear rate in the vein is

The coupled dynamics of $\{⟨\tau ⟩,{\tau }_s,⟨a⟩\}$ are now fully specified through Equations 1–3. To simplify our analysis, we now explore the reduced system $\{{\tau }_s,⟨a⟩\}$ by replacing $⟨\tau ⟩$ 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 $\{{\tau }_s,⟨a⟩\}$ reproduces the key features of the trajectories observed experimentally. We find two stable fixed points at (0, 0) and $({\tau }_0,{⟨a⟩}_{\mathrm{stable}}(R/R_{\mathrm{net}},{\tau }_0))$, and one unstable fixed point at $({\tau }_0,{⟨a⟩}_{\mathrm{unstable}}(R/R_{\mathrm{net}},{\tau }_0))$ (see Figure 3B). The stable fixed point with finite radius, $({\tau }_0,{⟨a⟩}_{\mathrm{stable}})$ 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 $\{{\tau }_s,⟨a⟩\}$ as those observed experimentally. Trajectories spiral in the clockwise direction near the stable fixed point $({\tau }_0,{⟨a⟩}_{\mathrm{stable}})$ (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 $⟨\tau ⟩$ are then closely related to that of ${\tau }_s$.

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 $R/R_{\mathrm{net}}$ and on the value of the target shear rate ${\tau }_0$ 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 ${\tau }_0$. The force balance derivation in Marbach et al., 2023 finds that the shear rate reference ${\tau }_0$ is related to the local fluid pressure $P$, as ${\displaystyle {\tau }_0\sim {\tau }_{\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}}-⟨P-P_0⟩/\mu }$ (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,${\displaystyle P_0}$ , namely the atmospheric pressure. We recall that $\mu$ is the fluid viscosity. Finally, ${\tau }_{\mathrm{active}}={\sigma }_{\mathrm{active}}/\mu$ is a shear rate related to the active stress ${\sigma }_{\mathrm{active}}$ 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 $R/R_{\mathrm{net}}$ but also ${\tau }_0$ is a flow-based parameter, integrating network architecture.

In our experimental full network samples, we can calculate both the relative resistance $R/R_{\mathrm{net}}$ 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 $\tau$ may suffer different fates, as described through Equation 1. This is a radical shift compared to previous theoretical works which consider that ${\tau }_0$ 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).

Figure 4

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The relative resistance $R/R_{\mathrm{net}}$ varies over orders of magnitude (Figure 4B), with values that are not correlated with vein size (see Appendix 1—figure 6). Rather, $R/R_{\mathrm{net}}$ 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 $R/R_{\mathrm{net}}$. 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 ${\displaystyle R\gg R_{\mathrm{n}\mathrm{e}\mathrm{t}}}$ (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 $R/R_{\mathrm{net}}$. In this latter case, all flow paths have to pass through small nearby veins and, hence, have high resistance ${\displaystyle R_{\mathrm{n}\mathrm{e}\mathrm{t}}\gg R}$ (see blue arrow in Figure 4B). $R/R_{\mathrm{net}}$, therefore, reflects the relative cost to transport fluid through an individual vein rather than through the rest of the network.

The relative resistance $R/R_{\mathrm{net}}$ 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, $Q^2{R}_{\mathrm{net}}/2$. Hence, we may expect that when in a given vein ${\displaystyle R>R_{\mathrm{n}\mathrm{e}\mathrm{t}}}$, it is energetically more favorable to flow fluid through the rest of the network and hence to shrink the vein. Reciprocally, if ${\displaystyle R<R_{\mathrm{n}\mathrm{e}\mathrm{t}}}$, we expect that the vein is stable. Analyzing our equations gives further support to this intuitive rule. When ${\displaystyle R\gg R_{\mathrm{n}\mathrm{e}\mathrm{t}}}$, from Equation 3, we may expect $⟨\tau ⟩$ to be relatively small, in particular, small relative to the vein’s specific steady state ${\tau }_0$ and hence via Equation 1 the vein would likely shrink. Reciprocally, if ${\displaystyle R\ll R_{\mathrm{n}\mathrm{e}\mathrm{t}}}$, we may expect $⟨\tau ⟩$ to be relatively large compared to its specific ${\tau }_0$, and hence the vein is stable. Yet, since $R/R_{\mathrm{net}}$ is nondimensional, it can provide more systematic insight than $⟨\tau ⟩$, since ${\tau }_0$ is not known a priori. Notice that the red arrow in Figure 4B presents a shrinking vein that indeed verifies ${\displaystyle R>R_{\mathrm{n}\mathrm{e}\mathrm{t}}}$. 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 $R/R_{\mathrm{net}}$, 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 $R/R_{\mathrm{net}}$ and the local pressure $P$ via the target shear rate, control vein dynamics on the basis of three key network topologies of a vein.

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.

Figure 5

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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 $R_{\mathrm{net}}$ cannot be calculated in a dangling end and cannot play a role. The shear rate in a dangling end is simply $⟨\tau ⟩=\frac{4⟨|Q_{\mathrm{in}}|⟩}{\pi {⟨a⟩}^3}\simeq \frac{32Lϵ}{⟨a⟩T}$. 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, ${\tau }_0$ determines the threshold for growth over shrinkage. Since ${\tau }_0\sim {\tau }_{\mathrm{active}}-⟨P-P_0⟩/\mu$ a large $⟨P⟩$ decreases ${\tau }_0$. 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.

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 R₁ and R₂. 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 ${\displaystyle R_1/R_{\mathrm{n}\mathrm{e}\mathrm{t},1}>R_2/R_{\mathrm{n}\mathrm{e}\mathrm{t},2}}$, 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 $R/R_{\mathrm{net}}$ in our full network (Figure 5B), we find that a vein with a large relative resistance ${\displaystyle R/R_{\mathrm{n}\mathrm{e}\mathrm{t}}>1}$ will vanish. In contrast, a nearby, nearly parallel vein with $R/R_{\mathrm{net}}\simeq 1$ will remain stable.

The relative resistance $R/R_{\mathrm{net}}$ 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 $R/R_{\mathrm{net}}$ over the shear rate itself: $R/R_{\mathrm{net}}$ is straightforward to compute from global network architecture as it does not require to resolve flows, and it is non-dimensional.

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 $R/R_{\mathrm{net}}$ 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.

Microscopic images of all the specimens used for this study are available as movies in MP4 format. Numerical data analysis available at https://doi.org/10.14459/2023mp1705720.

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.

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.

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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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 $t_{\mathrm{ave}}\simeq 2-3T$ ($T\simeq 120\mathrm{s}$). The $i^{\text{th}}$ element of the smoothed signal is given by ${\tilde{x}}_i=\frac{1}{N}{\sum }_j^N{x}_{i-\frac{N}{2}+j}$, where $N$ is the window size. 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 $a(t)\simeq ⟨a⟩(t)(1+ϵ\cos (2\pi t/T))$, where the average radius slowly evolves in time as $⟨a⟩=L\cos (2\pi t/t_{\mathrm{adapt}})$. The flow in the vein is $Q=L\frac{d(\pi a^2)}{dt}$ and therefore the shear rate at lowest order in $ϵ$ is

${\displaystyle \tau (t)=\frac{4}{\pi }\frac{|Q|}{a^3}\simeq \frac{4}{\pi }\frac{4\pi L}{T⟨a(t)⟩}ϵ|\sin (2\pi t/T)|.}$

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

${\displaystyle ⟨\tau (t)⟩=\frac{1}{T}{\int }_0^T\tau dt=\frac{2}{T}\frac{4}{\pi }\frac{4\pi L}{T⟨a⟩}ϵ{\int }_0^{T/2}\sin (2\pi t/T)=\frac{32L}{\pi T⟨a(t)⟩}ϵ}$

which is exactly the amplitude of the raw $\tau$ 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.

Appendix 1—figure 1

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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).

Appendix 1—figure 2

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Appendix 1—figure 3

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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 $R/R_{\mathrm{net}}$ in Appendix 1—figure 4E and Appendix 1—figure 5E. We find a number of veins with ${\displaystyle R/R_{\mathrm{n}\mathrm{e}\mathrm{t}}>1}$, 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 $R_{\mathrm{net}}$ (C) and $Q_{\mathrm{in}}=⟨\left|\pi \frac{d{a}^2}{dt}\right|⟩$ (F). We find that $Q_{\mathrm{in}}$ typically evolves like the vein radius: showing larger values (light blue) for larger veins and reciprocally smaller values (dark blue) for smaller veins. $R_{\mathrm{net}}$ in contrast evolves quite dramatically from vein to vein, according to how the vein is close or not to major highways.

We also provide cross-correlation data between specific quantities and initial vein radius $⟨a⟩$ at the beginning of the experiment, $R_{\mathrm{net}}$, $R/R_{\mathrm{net}}$, $Q_{\mathrm{in}}$ and $⟨P⟩$ (A,B,D,E). We find that the only quantity that is significantly correlated with $⟨a⟩$ is $Q_{\mathrm{in}}$, coherently since we expect $Q_{\mathrm{in}}\propto ⟨a^2⟩$.

Appendix 1—figure 4

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Appendix 1—figure 5

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Appendix 1—figure 6

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Appendix 1—table 1

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	–	–
$\mu$	Inner fluid viscosity	Long	Long
$\omega =2\pi /T$	Contraction frequency	–	–
$P$	Inner fluid pressure	Long	Short and long
${\displaystyle P_0}$	Atmospheric pressure	–	–
$Q$	Fluid flow pervading a vein	Short	Short and long
$Q_{\mathrm{in}}$	Fluid flow generated by peristaltic contractions in a vein	Short	Short and long
$Q_{\mathrm{net}}$	Fluid flow generated by the rest of the network	Short	Short and long
$R$	Resistance of a vein, $R=8\mu L/\pi a^4$	Short	Short and long
$R_{\mathrm{net}}$	Resistance of the rest of the network attached to a vein at both vein ends	Short	Short and long
$\tau$	Shear rate on a vein’s inner wall	Short	Short and long
${\tau }_s$	Sensed shear rate for adaptation	Short	Long
${\tau }_0$	Steady state shear rate, ${\tau }_0\sim {\tau }_{\mathrm{active}}-⟨P-P_0⟩/\mu$	Long	Long
${\tau }_{\mathrm{active}}$	Active contribution to ${\tau }_0$, due to energetic consumption from the actomyosin cortex to maintain contractions	Long	Long
$t$	Time	–	–
$t_{\mathrm{adapt}}\simeq 10-100\min$	Long-time adaptation timescale	Long	Long
$t_{\mathrm{delay}}\simeq 2\min$	Delay between adaptation and shear rate	Long	Long
$T\simeq 1-2\min$	Peristaltic contraction period	–	–
$⟨X⟩$	Long-time average of $X$	–	Long