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BIOPHYSICS AND PHYSICS COMPUTATIONAL BIOLOGY observations, we theoretically formulate a mechanism of encod- Initially, application of the stimulus causes a large-scale ing the location of a nutrient stimulus into the network hierar- increase in diameter of the tubes in immediate proximity to the chy: the stimulus triggers the release of a chemical agent that stimulus location and a decrease in tube volume farther away from causes the gel-like tube walls to soften, resulting in a significant the stimulus location (Fig. 2A). Close inspection of the stimulus- dilation of tubes receiving sufficient soluble agent. The chem- induced relative changes in tube diameter across the network ical agent propagates by contraction-driven cytoplasmic flows, reveals spatial heterogeneity in the response: the diameters of the initiating dilation downstream of the stimulus and thus, propa- thick tubes directed toward the nutrient stimulus increase, while gating the information about the stimulus location. Numerical the diameters of thinner tubes in general decrease—the more the solution of the theoretical model correctly predicts the exper- farther the distance from the stimulus or the farther from the thick imentally observed flow-dependent tube diameter response to tubes directed toward the stimulus location (Fig. 2B). Altogether, the stimulus—generating the new network hierarchy. Finally, this heterogeneous response increases the hierarchy in network we show that memories encoded in network hierarchy are read tube diameters (SI Appendix, Fig. S1). out as network morphology is shaping the direction of future The observation that thinner tubes that are close to thick tubes migration. This strategy to encode information by strengthening do not shrink compared with those farther away from thick tubes existing transport connections close to the stimulus reminisces correlates with the dispersion pattern of chemicals in networks of associative memory formation and thus may be important for (24) and thus, suggests that fluid flows-based transport is at the living flow networks in general. basis of the observed network reorganization.

Results Fluid Flow Propagates Stimulus into the Network. To investigate if Changes in Tube Diameter Hierarchy Imprint Nutrient Stimulus Posi- transport by flow is underlying the change in tube diameters, we tion. To study the response of P. polycephalum to nutrient stim- sort all network tubes by their Euclidean distance to the stim- uli, we collected time series of bright-field images of the organism ulus location and display their diameter dynamics (Fig. 3). We while foraging over approximately 5 h, during which a nutrient find that the dilation of tubes propagates in a wave-like manner stimulus was applied as a single local source in close proximity from the stimulus site at the speed of 15µm/s, corresponding to the network (Fig. 1). Within 45 min after stimulus application, to the speed of particles advected through the network, orders the organism internally reorganized to create a new migration of magnitude slower than change in pressure (21) yet outpacing direction facing the stimulus. Subsequently (i.e., 90 and 310 min), the speed by diffusion (SI Appendix, Fig. S2). The heterogeneous the organism migrated toward the stimulus. At 310 min, the spread of the dilation front further excludes diffusion as driver organism almost fully consumed the nutrient source and con- of transport (SI Appendix, Fig. S3). Direct observation of flow tinued to forage. Strikingly, a ring of thick tubes around the shear rate during changes in tube diameters also explicitly rules consumed nutrient source imprinted the stimulus location in the out shear forces as driver (SI Appendix, Fig. S4). network. This observation ignited the idea that tube diameters A remarkable feature of the observed process is the persis- may encode the location of a nutrient source. Given that P. poly- tence of tube dilation. The new distribution of tube diameters is cephalum is known to make decisions even within 10 to 20 min, established within about 15 min of stimulus application and per- we next focused on the immediate response after the stimulus sists to the end of the experiment (45 min). The stability of the was applied. new tube diameters indicates a lasting change of the mechanical To quantify network morphology and its dynamics in response properties of the tube material caused by the stimulus that would to nutrient stimuli, we trimmed the specimen and analyzed allow withstanding the rapid movement of mass by shuttle flows networks with a stable morphology (i.e., before the onset of throughout the network. The rapid and significant, up to twofold, migration and without overt reaction to microscope light). We dilation of tubes near the stimulus site suggests that a chemi- followed the initial reorganization during the 45 min after stimu- cal agent acts on the mechanical properties of the tubes. Direct lus application (Fig. 2). We subsequently analyzed the time series softening of tube wall would also account for the network-wide of bright-field images to quantify the dynamics of tube diame- transient increase in contraction amplitude (SI Appendix, Fig. ters. A tube designates a segment between two network vertices. S5). Agent-driven softening of tube walls is in line with observa- By diameter, we refer to the diameter trend of the rhythmi- tions of migration fronts being softer than the remaining tubular cally contracting tube in order to identify long-lasting changes network, arising from the difference in actomyosin organization in the network (Materials and Methods and SI Appendix). The (29–32). network rapidly acquires a new spatial distribution of tube diam- eters within about 15 min after stimulus application—a pattern Mechanism of Encoding Memory in Differential Tube Diameter that persists until the end of the experiment (45 min), when the Dynamics. Experiments suggest the following mechanism to organism starts to migrate toward the stimulus. encode memory of a stimulus location into network hierarchy.

min -60 min 0 min 45 min 90 min 310

5 mm 5 mm 5 mm 5 mm 5 mm

Fig. 1. Memory of a nutrient stimulus’ position is encoded in network hierarchy. Bright-field images of a foraging P. polycephalum network subject to a localized nutrient stimulus (red arrow) applied at 0 min. The network previously migrating to the right reorganizes migration direction facing the nutrient within 45 min. Subsequently, nutrient is exhausted (90 min) until foraging is resumed (310 min). Nutrient location is imprinted in the network hierarchy by thick tubes formed around the nutrient source—persisting long after the nutrient is consumed.

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BIOPHYSICS AND PHYSICS COMPUTATIONAL BIOLOGY increasing distances from the stimulus site (Fig. 4A). Shortly after sequent dilation farther from the stimulus. This tube behavior stimulus release close to tube end, the tube segment closest to is robust among all datasets (SI Appendix, Fig. S9–S13). Pre- the stimulus site dilates, while the other reference segment far- dicted dynamics along a single tube and experimentally observed ther away decreases in diameter—due to conservation of fluid dynamics within a network strongly resemble each other apart volume. Only as time progresses does the tube segment farther from two factors that arise due to the simplification of a single downstream receive softening agent by flow driving it to dilate tube in the model. First, the magnitude of dilation and shrink- subsequently (SI Appendix, Fig. S6). Yet, the tube segment down- age is directly tied to the total volume of the tube (37) and as stream does not dilate as much as the segment close to the such, smaller in the single tube compared with the voluminous stimulus site, thereby creating a hierarchy of tube diameters in network. Second, in a network, the softening agent is diluted in our beforehand uniform tube. The hierarchy arises as tube seg- the multitude of tubes: the farther it gets from the stimulus site, ments downstream receive less agent. These different softening the less softening agent gets to an individual tube. This results agent concentrations are not caused by agent’s decay, which is in an on average smaller dilation and subsequent tube shrinkage. −1 very slow kdeg = 0.001 s (here, also exemplified by the long- Taking these geometry-related considerations into account, the term persistence of the dilation). Rather, the flow velocity at resemblance of predicted dynamics and observations is striking. stimulus site is much slower than in the middle of the tube due to Finally, we categorize our experimental datasets by contrac- the eminent closed boundary at the end of the tube. Thus, most tion frequency (SI Appendix, Fig. S7) and quantify the response of the soluble agent stays close to its release site, while only a time across the networks (Fig. 4D). As predicted, the observed part gets advected, causing the delayed and decreased response response time increases as contraction frequency decreases— downstream. despite the very different morphologies and sizes of the net- We next aim to make a quantitative prediction from the model works. Taken together with the dynamics of tube diameters, this that exemplifies the crucial role of flow-based transport. To this agreement of data and prediction confirms that the advection of end, we probe how the tube diameter response time is altered a softening agent by fluid flows imprints nutrient location into a by changing flow velocities. Here, we use that contraction fre- specific change in hierarchy of tube diameters. quencies, which directly control flow velocities, vary naturally in P. polycephalum. We quantify as response time the time between Encoded Memory Is Read Out. Taken together, our experiments stimulus application and minimal diameter in our tube segment and model show that a nutrient stimulus changes the hierarchy farther along the tube (Fig. 4A) while decreasing contraction fre- of tube diameters and thereby, imprints its location into the net- quency (Fig. 4B). We find that the response time increases with work. To state that the imprint in network hierarchy poses a decreasing contraction frequency, underlining the central role of memory for the organism, we need to test if the memory (i.e., flow-based transport in setting up the hierarchy in response to network hierarchy) can be read out. Does the network respond the stimulus. differently to a new nutrient stimulus when previously having Having in hand these two predictions on the tube diameters encountered one in the same direction or not? dynamics close and far to the stimulus and the response time, Directly testing this is a futile approach as for P. polycephalum, we return to our experimental data. We automatically sort all there are never two identical networks to compare dynamics tubes in the network [N(tubes) = 2,165] by their response to the with and without memory. Furthermore, the innate tendency stimulus (described in SI Appendix), resulting in the two char- of the network to reorganize its topology and migrate becomes acteristic dynamics predicted by the model (Fig. 4C), with pure adverse to quantification beyond the time window used here. tube dilation close to the stimulus and tube shrinkage and sub- Fortunately, every network comes with a given hierarchy of

A B

C D

Fig. 4. Theoretical model of elastic tube dilation triggered by flow-transported softening agent captures characteristic tube dynamics observed in the experiment. (A) Predictions of a theoretical model of a closed peristaltic tube (sketch) on the tube diameter dynamics at the marked segments. Releasing a soluble wall-softening agent (red symbol/vertical line) within the elastic tube predicts characteristic tube dilation and relaxation. Tube dilation response time (dotted line) is shifted in time due to the agent being transported by peristaltic flow. (B) A decrease in contraction frequency slowing down flows directly increases the tube segment’s response time until dilation. (C) Experimental tube diameter dynamics are fully classified into the two characteristic dynamics predicted by the model (sketch). The dynamics of the mean tube diameter averaged over each ensemble of tubes follow theoretically predicted tube dynamics. N(tubes) = 2,165. (D) Quantification of the response time until dilation in experimental datasets shows increase of response time with decreasing contraction frequency, in line with theory. Error bars show the full range of tube response times in every dataset. N(tubes) = 2,165, 486, 438, and 45 for 10, 8, 6, and 4 mHz, respectively.

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BIOPHYSICS AND PHYSICS COMPUTATIONAL BIOLOGY plasmodia were imaged with a Zeiss Axio Zoom V.16 microscope equipped of 60.0 at 4 min. In space, the stimulus is represented as a narrow Gaus- with a Hamamatsu ORCA-Flash 4.0 digital camera and a Zeiss PlanNeoFluar sian concentration profile close to the left end of the tube. The parameters 1×/0.25 objective. The Zeiss Zen 2 (Blue Edition) software was used for used in the simulations are as follows: resting tube radius a0 = 50 µm, tube imaging. An image was acquired every 3 s. Stimuli were applied by placing length L = 0.5 cm, contraction frequency ω = 2π/120 s (32), wavelength small pellets of heat-killed HB101 bacteria on the agar using an Eppendorf of the traveling wave λ = L, dynamic viscosity of cytoplasm µ = 6.4 · 10−3 2 Microloader pipette tip. In two experiments presented in Movies S3 and S5, Ns/m (49), tube wall height h = 0.1a0, amplitude of contraction-inducing 1 µL of a solution containing 0.5 M glucose and 0.1 M leucine was placed on cortex tension A = 10 Pa (21), molecular diffusivity of small chemical species −10 2 −1 the agar close to the network as a nutrient stimulus. κ = 10 m /s, and decay rate of the chemical agent kdeg = 0.001 s . The parameters give rise to Reynolds number Re = 2ua¯ 0/ν ≈ 0.001. The effec- Image Analysis. First, the network skeletons, pixel intensities, and tube tive resting elastic modulus is E = 10 Pa from Young’s modulus of 100 Pa (20, radii dynamics were extracted using a custom-written MATLAB (The Math- 50), and for the feedback of chemical agent on cortex mechanics, δE = 2.5Pa works) code (22). Subsequently, tube diameters are calculated from the and hc0i = 10.0 were chosen to result in E = 0.8E0 at the minimum. time average of tube radii over 100 frames, thereby averaging out radii contractions. Data Availability. All study data are included in the article and/or Support- ing Information. Numerical Simulations. The equations of the theoretical model were solved numerically in a custom-written MATLAB (The Mathworks) code using a ACKNOWLEDGMENTS. We thank Natalie Andrew for Movie S3. M.K. theta-weighted Crank–Nicholson scheme with closed boundary conditions. acknowledges the support of International Max Planck Research School for Initially, the tube has a uniform radius a0. The softening agent is introduced Physics of Biological and Complex Systems. This work was supported by the at stimulus time, growing exponentially in time to a final concentration Max Planck Society.

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