Vein fate determined by flow-based but time-delayed integration of network architecture
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Evaluation Summary:
Fluid flows in networks are ubiquitous, and in many living systems the networks are not static but instead can rearrange over time. Using vascular networks formed by the slime mold Physarum polycephalum, Marcbach et al. demonstrate that there is a time delay between the change in the flow and the change in the network geometry. They present a mechanical model of vein-radius adaptation leveraging the negative normal stress response of the actin cytoskeletal network lining the vein walls. More generally, the authors make use of the unique advantage of this simple model vascular system to connect the local shear rate to the network reorganisation and how it depends on its architecture. There are features to their work that are new to the literature and that can be impactful in advancing the field.
(This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. The reviewers remained anonymous to the authors.)
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Abstract
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.
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Evaluation Summary:
Fluid flows in networks are ubiquitous, and in many living systems the networks are not static but instead can rearrange over time. Using vascular networks formed by the slime mold Physarum polycephalum, Marcbach et al. demonstrate that there is a time delay between the change in the flow and the change in the network geometry. They present a mechanical model of vein-radius adaptation leveraging the negative normal stress response of the actin cytoskeletal network lining the vein walls. More generally, the authors make use of the unique advantage of this simple model vascular system to connect the local shear rate to the network reorganisation and how it depends on its architecture. There are features to their work that are new to the literature and that can be impactful in advancing the field.
(This preprint has …
Evaluation Summary:
Fluid flows in networks are ubiquitous, and in many living systems the networks are not static but instead can rearrange over time. Using vascular networks formed by the slime mold Physarum polycephalum, Marcbach et al. demonstrate that there is a time delay between the change in the flow and the change in the network geometry. They present a mechanical model of vein-radius adaptation leveraging the negative normal stress response of the actin cytoskeletal network lining the vein walls. More generally, the authors make use of the unique advantage of this simple model vascular system to connect the local shear rate to the network reorganisation and how it depends on its architecture. There are features to their work that are new to the literature and that can be impactful in advancing the field.
(This preprint has been reviewed by eLife. We include the public reviews from the reviewers here; the authors also receive private feedback with suggested changes to the manuscript. The reviewers remained anonymous to the authors.)
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Reviewer #1 (Public Review):
The authors report experiments and a mathematical model to understand how a flow network of Physarum polycephalum rearranges the channel radii in time. The topic is interesting since fluid flows in networks are ubiquitous and in many living systems the networks are not static but instead can rearrange over time. The variables that control the rearrangements, including growing and shrinking different flow channels, are still not understood though apparently often it is assumed that the local shear rate dictates time-dependent network dynamics. In this paper, the authors demonstrate using experiments that there is a time delay between the change in the flow and the change in the network geometry, and that network architecture-dependent parameters, such as the local shear rate in a channel, and the resistance …
Reviewer #1 (Public Review):
The authors report experiments and a mathematical model to understand how a flow network of Physarum polycephalum rearranges the channel radii in time. The topic is interesting since fluid flows in networks are ubiquitous and in many living systems the networks are not static but instead can rearrange over time. The variables that control the rearrangements, including growing and shrinking different flow channels, are still not understood though apparently often it is assumed that the local shear rate dictates time-dependent network dynamics. In this paper, the authors demonstrate using experiments that there is a time delay between the change in the flow and the change in the network geometry, and that network architecture-dependent parameters, such as the local shear rate in a channel, and the resistance of flow in a part of the network, relative to the resistance in the rest of the network, can be used to predict vein dynamics. For example, the authors observe vein dynamics by tracking vein radius and shear rate over time and identify regular behavior, e.g., usually stable veins perform looping trajectories in the shear rate-radius space shown in Figure 1, which appears to correlate with an in/decrease in shear followed by an in/decrease in vein radius yielding shear feedback on local vein adaptation. In contrast, usually in shrinking veins, the relation between shear and vein adaption is ambiguous, to use the authors' words. Their data makes clear these main features and the authors construct a mathematical model that helps understand the observed instabilities (channels shrink and disappear) or stability (channels can periodically grow and shrink). It is the features of the time dependence of the network, and identifying variables and a macroscopic model for the dynamics, that I think are the novelties of the paper and so most likely to be impactful in the field, e.g., vein fate being determined by network architecture dependent parameters, such as relative pressure and relative resistance. That said, I find some of the writing unclear and some of the figures challenging to read and understand. Also, it was unclear what might have been reported in several of the referenced papers that highlight dynamical features relevant to this paper.
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Reviewer #2 (Public Review):
This paper presents new experiments and a theoretical model that addresses dynamical aspects of vascular adaptation in the slime mold Physarum polycephalum. This is perhaps the simplest organism in which the celebrated law for vascular morphology by Murray (1926) has been demonstrated, and hence insights from this experimentally amenable system may reveal design principles relevant to vasculature adaptation in many other systems. Combining high-resolution measurements of flow inside individual veins of the network with time-lapse imaging of the entire vascular network's morphology, the authors discovered that the radius of veins changes in a manner that is correlated with changes in the local shear rate of fluid flow, but with a time delay.
Based on these observations, the authors developed a detailed …
Reviewer #2 (Public Review):
This paper presents new experiments and a theoretical model that addresses dynamical aspects of vascular adaptation in the slime mold Physarum polycephalum. This is perhaps the simplest organism in which the celebrated law for vascular morphology by Murray (1926) has been demonstrated, and hence insights from this experimentally amenable system may reveal design principles relevant to vasculature adaptation in many other systems. Combining high-resolution measurements of flow inside individual veins of the network with time-lapse imaging of the entire vascular network's morphology, the authors discovered that the radius of veins changes in a manner that is correlated with changes in the local shear rate of fluid flow, but with a time delay.
Based on these observations, the authors developed a detailed theoretical framework to explain some of the non-trivial dynamics observed at the network level in the experiments. This framework consists of three parts: (i) a model of vein radius adaptation in response to changes in shear rate, which takes into account force balance in the vein walls and the experimentally observed time delay in that response, (ii) a fluid mechanical model of flows in contractile veins, and (iii) application of Kirchoff's circuit laws within the measured network morphology, to extract network-dependent parameters required to apply the flow and vein adaptation models (I & ii) to the experimental data. From a theoretical point of view, the main novelty here is (i), which develops a mechanistic picture of vein radius adaptation based on shear-rate feedback, that is, how vein radius changes in response to the sensed shear rate until some target shear rate is reached. From a functional perspective, shear-rate feedback ensures Murray's-law scaling of vein radii with the volumetric flow.
A key mechanical ingredient of the author's new model is a peculiar feature of actin cytoskeletal networks, which are known to drive the contractile dynamics of veins in P. polycephalum. Specifically, networks of semi-flexible polymers such as actin have been shown to demonstrate negative normal stresses in response to shear. The authors argue that this property can provide a mechanism for the flow-induced tangential shear on the actin network to drive a dilation of the vein radius. The balance of the resulting normal stress with other forces acting at the wall, such as active contractility and friction in the actin network and turgor pressure, then determines both the adapted-state shear rate and the adaptation timescale. Although this force-balance model does not on its own account for the experimentally observed time delay of the radius changes upon a change in shear rate, adding an additional phenomenological equation to account for that time delay, together with Kirchoff's circuit laws (iii) yields a dynamical system describing vein radius and shear rate changes for a given circuit configuration. The authors show that these dynamical system models can successfully fit a variety of observed patterns of vein radius changes, taking the measured network morphology and local shear rate as input, and with just two free parameters (the adaptation timescale and target shear rate).
Although a number of features of the model (for example, the mechanistic cause of the time delay, and various assumptions of the force-balance model) remain open for future investigation, this theoretical framework provides a concrete and falsifiable physical model of vein radius adaptation dynamics in P. polycephalum that satisfies Murray's law and hence should be of interest to a broad range of readers interested in vascular adaptation.
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