Active perception during angiogenesis: filopodia speed up Notch selection of tip cells in silico and in vivo

To quantify this difference in spatial periodicity, we employed the Lomb-Scargle Periodogram—which is a commonly used signal processing algorithm used to assess the presence of periodic modes [74]. This algorithm works essentially by checking how well sine waves of a given frequency are capable of describing the data, creating a ‘spectrum’ of values to evaluate the fit through a range of frequencies (figure 2f, right subpanel). In the control case, we find a sharp peak in the frequency spectrum at approximately 0.07 Hz, and a significantly smaller peak in the area around approximately 0.02 Hz. The 0.07 Hz mode indicates the presence of periodic peaks of period P=(2π/f)≈ 90 length units in pErk activations: a regular spacing that corresponds to the spacing of the ISVs. On the other hand, the 0.02 Hz peak corresponds roughly to a spatial period that spans our entire length axis and thus is merely the negligible equivalent of a ‘constant’ offset term in the sinusoidal fit. In the LatB condition, the 0.07 Hz peak loses more than half of its amplitude—indicating that a large part of the periodicity is not present, and the pattern is markedly more diffuse.

Hence, in vivo observations were consistent with simulation predictions that filopodia speed up the collective decision-making process. Likewise, tip cells were slower to emerge from the DA and lead ISV branches (figure 2f), in keeping with the time ordering of events predicted in the simulations, whereby (i) Vegfr is initially activated in many cells in response to Vegf before (ii) slower commitment to tip identity and behaviour defined by filopodia-regulated Dll4-Notch lateral inhibition. Hence, filopodia may influence the speed of tip cell selection to ensure blood vessel branching occurs within the precise time constraints of the developing tissue.

The ability to dynamically select between two distinct phenotypes while intermediate states are unstable is a hallmark of bistable regulatory systems. In regulatory circuits, bistability is a mechanism well documented to enable continuous systems to act like discrete switches, by destabilizing possible intermediate values of the relevant state parameters in all but the two stable configurations [75–78]. Bistable systems are robust because small perturbations around one of their steady states simply lead to the system returning to its original configuration due to the inherent dynamics involved.

In a vessel comprised ECs with bistable tip/stalk properties, the attractiveness of choosing one state or the other would enable them to come to a collective patterning decision faster by avoiding drawn out battles in intermediary states. We recently identified that positive feedback to the Vegf sensory system by TM4SF18 slowed selection with indications that this was due to bistable properties [20]. Hence, we seek here to evaluate the hypothesis that positive feedback specifically caused by filopodia movements up a gradient of Vegf speed up selection by conferring bistable decision dynamics to individual cells using a specifically adapted MSM in silico modelling set-up, building on preliminary studies [79].

To explain our approach and expected results when assessing this hypothesis, we will use a simple mechanical analogy of a bistable system. The analogy is of a metallic ball found on a double-well landscape (figure 3a). In this case, the physical shape of the landscape supporting the ball against gravity creates two stable (attractor) states, i.e. the bottom of each well. In the presence of some small noise able to overcome locally stable bumps in the surface, this mechanical system will not linger in an intermediate state at the top of the barrier or on its slope. Transitions between the two locally stable states take place abruptly, whenever forces on the ball exceed the potential barrier imposed by the presence of gravity. Moreover, slowly tuning the environment by moving a magnet positioned underneath the track left to right and back again (such that the ball is restricted to the left/right/left valleys) uncovers a signature behaviour of bistable systems, namely hysteresis (figure 3b). Charting the position of the ball as a function of the position of the magnet clearly reveals the initial-condition dependence of the ball's response. For any position of the magnet within a window around the barrier, the ball can settle into either of the two valleys, but which one depends on its recent history of positions (figure 3c). It cannot be stable in the transition state on top of the barrier; it will fall into one of the valleys. In a system without hysteresis, such as for example if the landscape was completely flat (figure 3b), the ball's resting position would be independent of its past values (depending only on the instantaneous value of the magnet's position) and two curves would be perfectly overlapping each other (figure 3d).

Our hypothesis, framed this way, would implicate filopodia in the generation of the hill in the middle of the hypothetical ‘landscape’ in this analogy (figure 3e,f,g). Figure 3. Hysteresis and bistability. (a) Metal ball on a double-welled landscape driven by a magnet: analogy for a bistable system, see text for explanation. (b) Metal ball on a flat landscape driven by a magnet: analogy for system that has lost bistability, see text for explanation. (c) Example bistable plot with hysteresis, showing the response variable ball position (BP) as a function of the driving variable magnet position (MP). Of note is the ‘gap’ between the dynamics of BP as MP is moved left (blue line) and the dynamics of BP as MP is moved right (red line)—a typical characteristic of systems with hysteresis. (d) BP dynamics as a function of MP in an example scenario where there is no bistability or hysteresis. Note how the state variable BP linearly responds to the driving variable MP. (e) When this analogy is applied to Notch selection in ECs, the two stable valleys in the landscape correspond to the tip and stalk states, and the barrier between them is hypothesized to be caused by filopodial activity. The external magnetic driving force is analogous to the external Dll4 received by the cell from its neighbours. (f) In the case of normal filopodia, the landscape is hypothesized to have a double well. In this case, we expect the dynamics to resemble those in (c). (g) In the case of reduced filopodial activity (LOF), the landscape is hypothesized to flatten. In this case, we expect the dynamics to resemble those in (d). (Online version in colour.)

To assess whether filopodia confer the bistable property of hysteresis to individual EC decisions during selection, we established a modified version of the tip cell selection MSM designed to perform hysteresis analysis on a single EC (see Methods—MSM hysteresis). One crucial difference to the model here is that though we simulate three cells, we only test the dynamical properties of the single central cell. The two ‘cells’ either side of the central EC in the model do not act or respond as ECs, but merely provide fixed sources of external Dll4 that the central cell can bind (Dext) akin to the magnet in the mechanical analogy (figure 4a). When the side cells have a fixed high level of Dll4, the central cell becomes inhibited via this Notch-Dll4 signalling prohibiting it from becoming a tip cell (figure 4b). Conversely when the side cells are fixed to present low Dll4, the middle cell receives no Notch signal and extends filopodia, becoming a selected tip (figure 4c). To assess hysteresis in simulations, Dext starts at zero and is then increased in small increments up to Dmax then decremented back down to zero, akin to the magnet movement left and right. Figure 4. Filopodia movement in Vegf gradients confers bistable properties to Notch patterning. (a) Schematic of the MemAgent-Spring Model Hysteresis (MSM-Hyst) testbed, designed to probe the behaviour of the single endothelial cell in the middle (through Dcell dynamics) in response to externally driven inhibitory Dext signals from two neighbouring pseudo-cells and VEGF in the environment. (b,c) Screenshots of the MSM-Hyst simulation testbed when Dext is at max (b) and min (c) levels. In (b) the central cell is inhibited, in (c) the central cell is uninhibited so has high Dll4 and filopodia (tip cell selected). Dll4 high cells, light green; Dll4 low cells. purple; VEGF gradient in the environment, dark green.

Note the peripheral pseudo-cells do not have filopodia, as they only provide Dext, they do not respond to Vegf or Notch themselves, in this highly constrained testbed. (d) Dcell dynamics as a function of Dext in cases where we covary the model parameters determining the steepness of the Vegf gradient (0.02 (LOF), 0.04 (control)), and the responsiveness (in terms of probability to extend) of the filopodia to sensed Vegf (F = 0.5 (LOF), 2 (control), 3 (GOF)). We can see that some parameter settings (e.g. F = 2, Vegf gradient = 0.04) yield bistable dynamics with hysteresis, while others (e.g. F = 2, Vegf gradient = 0.02) do not. N = 100. Shading indicates 1 standard deviation. (e) A cartoon demonstrating how the filopodia guidance parameter (G) would influence the behaviour of a growing filopodium in a two-dimensional schematic (the MSM simulation extends this to three dimensions). The central square represents the memAgent at the head of the filopodium, who must decide which direction to step every time step that the filopodium extends (see Methods for full details). G determines the probability that the memAgent will choose the adjacent square with the highest level of Vegf (Vegf high, pink; Vegf low, blue). Thickness of the arrows represents the probability of stepping into a given square. When G = 1 (left panel), there is a 100% probability of stepping into the pinkest square. As G is decreased, the probability of stepping into a square at random increases—represented by uniformly distributed arrows and a thinner arrow pointing to the pinkest square. Note that the probability of stepping into the pinkest square is never zero, as stepping at random might still land on it. (f) Dcell dynamics as a function of Dext where we vary the guidance parameter G. As G approaches 0, the hysteresis window shrinks and the bistable dynamics disappear.

For all plots, N = 100, shading indicates 1 standard deviation. F = 2 and V = 0.04 (both control settings). (Online version in colour.)

To evaluate the hypothesis that filopodia moving through a Vegf gradient generates positive feedback in the Vegfr system and confers bistability, we disaggregated that idea into parts and performed simulations varying each part. First, we covaried the probability of filopodia extension (F) with the slope of the linear Vegf gradient (V) and analysed the Dll4 dynamics of the middle cell (Dcell) as Dext levels were incremented up and down. In the ‘normal Vegf gradient’ (V = 0.04) with standard filopodia responsiveness (F = 2) both calibrated previously and used throughout MSM studies [24]) a bistable window is indeed apparent (figure 4d, control Vegf).

When F is reduced under normal environmental conditions (F = 0.5, V = 0.04), the loss in filopodia movement eliminates the bistable properties of the MSM-Hyst cell (figure 4d). Similarly, when the steepness of the Vegf gradient is halved (V = 0.02, LOF), the bistability is lost (F = 0.5,2) —though the hysteresis is partially rescued by GOF in filopodia movement (F = 3) (figure 4d). In cases with high filopodia activity and positive feedback from a steep Vegf gradient, such as F = 3, V = 0.04, the bistable window remains intact.

As a reduction in Vegf gradient steepness incurs a knock-on reduction to general Vegf concentration in this model (see Methods—MSM), this could be causing the very low Dcell values and a loss of bistability simply due to insufficient Vegf to initially activate the cells in the Vegf gradient LOF condition (figure 3d). To control for this, we used a previously calibrated version of the model with no gradient (a uniform distribution of Vegf) but using a concentration high enough to stimulate cells to achieve a pattern. Using our precursor memAgent model, we previously reported a slower time to select when Vegf is distributed uniformly compared to in a gradient [24] and indeed we see bistability is lost in these conditions in the MSM regardless of the level of F and the concentration of Vegf in the environment (electronic supplementary material, figure S2).

Taken together these simulations predict that positive feedback to VEGF sensing by means of filopodia movements through a Vegf gradient do confer rapid and robust bistable switch-like properties to single ECs experiencing changing Dll4 conditions from their neighbour cells.

Next, to test the importance of guidance up the Vegf gradient, as opposed to merely Vegf levels and filopodia responsiveness, we uncoupled the intrinsic feedback in the MSM between Vegfr (sensing) and filopodia extension (movement). We did so by modulating the guidance parameter G, which denotes the probability that a moving filopodium will choose to extend in the direction up the gradient—else taking a random step (figure 4e). G ranges from 0 to 1, with its default value set to 0.9 based on experimental observations [43].

By varying the guidance parameter (G), we found bistability properties are visibly disrupted with lower values, i.e. when the movement is less coupled to sensing (figure 4f, G = 0.1) at which point the active state loses stability. Though this low threshold value appears to minimize the importance of guidance in the hysteresis properties, the simplicity of the linear vertical gradient in the square lattice-based model means that a filopodium would step in the correct direction of positive feedback approximately 1/3 of the time even if it is moving completely at random. When adjusted for this fact, we see that even an approximately 50% reduction in effective guidance (from approximately 90% adjusted to 40% adjusted) is sufficient to diminish bistability. Conversely, GOF had no significant effects, but given that G is normally close to its maximum value this is to be expected.

In summary, altering parameters involving the guidance (G), filopodial responsiveness (F) or the gradient itself (V) can cause an abrupt transition to non-bistable tip/stalk dynamics. The bistability in this system is contingent on all three components being present—filopodia movement, the presence of a gradient, and the ability of the filopodia to navigate up the gradient.

Our results so far indicate (i) a role for filopodia in speeding up collective tip cell selection in vivo (figure 2) and (ii) that positive feedback between sensing and movement via filopodia confer bistable switch-like properties to facilitate rapid decision-making (figures 3 and 4). Here we ask the question are filopodia, therefore, a form of active perception? Are they akin to a basal cognitive function exploiting sensorimotor coordination to enable endothelial cells to adapt their function to the changing local environmental conditions by making ‘better’ (faster, more efficient) decisions?

To answer this question, we adopt the formalism of Lungarella, Sporns and Pegors who define active perception as the enrichment of the statistical structure of an individual's sensory input resulting from coordinated sensory and motor activity [64,80,81]. To detect active perception, they suggest a series of measures to probe statistical structure in a compressed representation of sensory input (the ‘Sensory Map’) induced by coordinated sensorimotor activity. They found that the measures they used have distinctive signatures, such as reduced Shannon entropy when their model systems are allowed to coordinate sensory and motor activity [64].

In order to apply this approach to ECs, we first established a simple two-cell version of the in silico MSM, as this is the minimal number of cells required to achieve the patterning outcome, and thus the simplest first case to delineate AP in a collective decision (figure 5a). We here identify the cells reaching a ‘pattern’ by comparing their Dll4 levels in order to pinpoint the earliest timepoint that Notch selection has stably occurred (in contrast to the longer-term Vegfr and filopodia dependent ‘time to pattern’ score used previously). A pattern is recorded when one of the two cells has obtained and maintained a substantial lead in Dll4 for a long enough time (see Methods—MSM hysteresis). The cells were again fixed in space and only able to extend and retract filopodia into a VEGF gradient, as per all simulations in this study. Figure 5. Signatures of active perception. (a) Two-cell MSM simulation set-up used for active perception signature studies. Sections of cell membrane (memAgents) with higher Vegfr activation, pink cubes; memAgents with low Vegfr activation, blue cubes. Structures extending out are filopodia. White cubes are memAgents on a junction with the other cell (indicating periodic boundary conditions far left and right). (b) Cartoon diagram of a sensory map. A sensory map, in this study, is a coarse-grained representation of what is sensed by endothelial cells. Activations of the Vegf receptors on memAgents throughout three-dimensional space (e.g. on filopodia) are mapped down onto the cell surface. All AP measure analysis is performed on this fixed spatial representation of Vegfr activations changing over time. (c) Example sensory maps with high and low entropy Vegfr activations (pink, high activation).

Because the sensory map itself is the surface of a cylinder (due to the shape of the cell surface in simulations), we may imagine it ‘unrolled’ as a two-dimensional rectangle, having various Vegfr activations across it. In the sensory map labelled ‘lower entropy’, the Vegfr activations are highly spatially concentrated in comparison to the ‘higher entropy’ case where they are more diffuse—yet their total intensity is the same. (d) Active perception measures as a function of time for different values of F, the filopodial responsivity parameter. Red lines are the mean values of N = 100 runs of the 2-cell MSM. Grey shaded area is the entire span of the data set. Vertical black line is the mean time at which the salt and pepper pattern was achieved (counting only the cases where a pattern was achieved within the max time steps, up to N = 100), vertical blue lines span the entire range of timing values. (e) Plots showing time evolution of measures for different values of G, the parameter determining filopodial guidance. (Online version in colour.)

We define our sensory map to be the projection of all activations of the Vegfr-2 receptors of both cells onto the fixed cylindrical vessel surface (figure 5b). Receptor activations occurring on a filopodium are projected from their point in three-dimensional space down to its nearest point on the cylindrical shell; this mapping preserves directional information (as opposed to collapsing activations down to the filopodium's base position on the cell surface) and results in a fixed spatial representation of the sensory input that the cells experience, overcoming the challenge of measurements on a dynamically changing form. The sensory map is considered at the scale of the vessel rather than the individual cell, i.e. we are specifically asking the question here does the vessel employ AP to solve a collective cell decision problem? Though this representation necessarily loses some information, we hypothesized the sensory map would retain enough directional information to detect a difference in the measures. This approach considers the question of AP at the scale of the vessel rather than the individual cell.

To search for signatures of active perception, we calculate the following measures across the sensory map at each time step in the simulation:

(1) Intensity (I): the sum of Vegfr activations (Vegfr*) across the sensory map with dimension N. This value is in units of Vegfr activations.I= ∑i=0N⁡Vegfr i∗

(2) Shannon Information Entropy (H): of the spatial distribution of Vegfr activations, for which we infer a distribution by binning with Laplace smoothing [82]. This is in units of Bits (basic unit of information, here quantifying uncertainty).H= −∑i=0N⁡pi(Vegfr∗)log2⁡( pi(Vegfr∗))Where pi is the inferred probability density of the amount of Vegfr activations at the ith site of the sensory map.

Rather than extensively testing for any possible structure in the sensory map, we chose these AP signature measures in particular because changes in these would be straightforward to interpret yet form a basis and approach easily extended to other measures in the future. Shannon Entropy (H) has a particularly useful property—it increases for broader distributions. For example, a sharply peaked Gaussian distribution will have a lower entropy than a uniform distribution over the same space. Although it does not strictly model coherence of values, we can use H as a proxy to describe the spatial spread of activating receptors on the cell surface (figure 5c). Crucially, this corresponds to the goal of the tissue as a whole to adopt an inhomogeneous pattern. As the entropy of activations decreases, the two-cell system gets pushed towards the state of one cell being a tip cell and receiving all of the sensory input and the other being a stalk cell and receiving none of it. Therefore, H may be interpreted as a quantitative readout of progress towards the goal of inhomogeneous cell state caused by unequal sensing.

We hypothesized that signatures of active perception would be apparent in the time evolution of the measures applied to the sensory map. For example, a reduction in the entropy of the sensory map over time would mean the driving Vegf signal that the vessel is receiving is being biased to be less spatially homogeneous across it. In the beginning, both cells have an equal chance of ‘winning’ and becoming the tip cell (this is due to stochasticity in the simulated filopodia extension process). Ultimately, the cell that wins is determined by the sensory input that the pair receive. If one cell starts to get more of the growth signal, this would both facilitate the production of an inhomogeneous pattern (e.g. more sensory input localized on one region of the vessel—the winning cell) and correspondingly reduce the measured entropy of the system's sensory map. If active perception is driving this entropy decrease, then the disruption of active perception would reduce or remove the reduction in entropy.

In terms of the intensity of Vegf activations across the vessel, we expected that the intensity as a function of time takes on a different form, for example by switching from a linear to a nonlinear regime due to increased feedback between movement and sensing. Overall, we expect a filopodia (LOF motor response) in a low Vegf gradient (LOF stimulus) would have a reduced AP ability to one with a stronger coupling between sensing the Vegf (V) and the motor response to extend filopodia (F) (i.e. F and V normal or GOF).

As before, we varied the probability of extending filopodia (via F) and the steepness of the Vegf gradient/concentration (V) to see how these measures change. There are marked differences between the sensing and motor response LOF and GOF cases, as shown by the behaviour of the measures as a function of time (figure 5d). In the normal filopodia conditions, entropy appears to decline slightly before a stable pattern is achieved—and then goes down by approximately 1 bit after the patterning is achieved. The fact that entropy decreases before the patterning is detectable in Dll4 levels, however, may be susceptible to the choice of pattern Dll4 criterion parameter (see Methods—AP signatures). Intensity takes a somewhat logistic-like shape at later times (figure 5d) after the initial phase of small fluctuations. In the cases where filopodia (actions) are inhibited (F LOF), entropy declines less sharply or not at all and the logistic-like shape of the intensity starts to flatten and ultimately disappears (figure 5d). In the GOF condition of F, we find that the cells undergo large fluctuations in Vegfr activations, which slow down patterning. This implicates filopodia in the generation of these patterns in the sensory map.

The large fluctuations in the GOF conditions of the F parameter are biologically accurate and reflect the behaviour of endothelial cells when they are exposed to too much Vegf. Fluctuations are explainable as a consequence of coupled negative and positive feedback mechanisms. This is a well-studied outcome in angiogenesis and has been linked to the malformation of vasculature in environments with high Vegf, such as those surrounding tumours [36]. The fact that alterations in AP-related features of the model can bring the system into and out of this regime highlights the promise of AP machinery as a potential therapeutic target [4].

A reduction in entropy indicates that the amplification of differences in the sensed Vegf signal have begun. A reduction in spatial homogeneity of the driving sensory signals is evident before the patterning decision at the level of Notch signalling is completed (figure 5d), thus sensory map structuring is predicted to bias one cell to win before lateral inhibition commits this choice. A sharper reduction in entropy corresponds to a faster patterning decision.