Mechanism of signal propagation in Physarum polycephalum

Karen Alima,b,1, Natalie Andrewa,b, Anne Pringlec,d, and Michael P. Brennera

aThe Kavli Institute for Bionano Science and Technology, Harvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138; bMax Planck Institute for Dynamics and Self-Organization, 37077 Goettingen, Germany; cDepartment of Botany, University of Wisconsin–Madison, Madison, WI 53706; and dDepartment of Bacteriology, University of Wisconsin–Madison, Madison, WI 53706

Edited by Nigel Goldenfeld, University of Illinois at Urbana–Champaign, Urbana, IL, and approved April 3, 2017 (received for review November 1, 2016) Complex behaviors are typically associated with animals, but the ter. Tubes are made of a gel-like outer layer and interior cyto- capacity to integrate information and function as a coordinated plasmic fluids. The outer layer houses an actin–myosin cytoskele- individual is also a ubiquitous but poorly understood feature of ton and the cytoskeleton generates periodic contractions of tube organisms such as slime molds and fungi. Plasmodial slime molds walls. Contraction amplitude and frequency generally increase grow as networks and use flexible, undifferentiated body plans to or decrease organism-wide when encountering an attractant or forage for food. How an individual communicates across its net- repellant, respectively (8–11). Contractions drive periodic cyto- work remains a puzzle, but Physarum polycephalum has emerged plasmic fluid flows, and these extend across an entire individual. as a novel model used to explore emergent dynamics. Within Intriguingly, the fluid flows are highly coordinated (12), and the P. polycephalum, cytoplasm is shuttled in a peristaltic wave driven phase of oscillations is tuned such that there is exactly one wave- by cross-sectional contractions of tubes. We first track P. poly- length across an individual, regardless of an individual’s size. Data cephalum’s response to a localized nutrient stimulus and observe suggest P. polycephalum is somehow able to measure its size. a front of increased contraction. The front propagates with a Although data also suggest P. polycephalum can communi- velocity comparable to the flow-driven dispersion of particles. cate across its entire body (13), we have no knowledge of the We build a mathematical model based on these data and in the nature of communication. Signals may propagate via elastic waves aggregate experiments and model identify the mechanism of sig- (14) or an advected molecular stimulus (15, 16) or electrical nal propagation across a body: The nutrient stimulus triggers impulses (17). Our very recent work tentatively suggests the sec- the release of a signaling molecule. The molecule is advected by ond hypothesis; flows generated from the coordinated contrac- fluid flows but simultaneously hijacks flow generation by causing tions of tube walls are used to increase the effective dispersion local increases in contraction amplitude as it travels. The molecule of molecules substantially beyond their pure molecular diffusiv- is initiating a feedback loop to enable its own movement. This ity, a phenomenon known as Taylor dispersion (18). However, mechanism explains previously puzzling phenomena, including these different possibilities can be definitively distinguished by the adaptation of the peristaltic wave to organism size and their velocities for signal propagation, which differ dramatically, P. polycephalum’s ability to find the shortest route between food and recognition of this fact opens up the possibility of understand- sources. A simple feedback seems to give rise to P. polycephalum’s ing communication, the key to understanding behaviors. complex behaviors, and the same mechanism is likely to function Building on our previous observations (12, 18), we now report in the thousands of additional species with similar behaviors. and characterize the mechanism of communication in P. poly- cephalum and demonstrate that a simple feedback between a acellular slime mold | transport network | behavior | Taylor dispersion signaling molecule and a propagating contraction front is suffi- cient to explain P. polycephalum’s sophisticated behaviors. The key experiment demonstrates that a localized nutrient stimulus ne of the great challenges of unraveling biological complex- Oity is understanding what kind of and how much compu- tational power is required for an organism to generate sophis- Significance ticated behaviors. Behaviors are typically associated with a nervous system, but many organisms without nervous systems How do apparently simple organisms coordinate sophisticated integrate information and function as coordinated individuals behaviors? The slime mold Physarum polycephalum solves (1); examples range from the ability of Escherichia coli to move complex problems, for example finding the shortest route up chemical gradients (2) to the ability of a multicellular fungus between food sources, despite growing as a single cell and to sense and precisely explore unoccupied space (3). A recently the lack of any neural circuitry. By carefully observing P. poly- published and striking example of a complex behavior involves cephalum’s response to a nutrient stimulus and using the data bacteria within a biofilm: When a Bacillus subtilis biofilm is to develop a mathematical model, we identify a simple mech- deprived of nutrients, bacteria are able to grow networks of chan- anism underpinning the slime mold’s behaviors: A stimulus nels and evaporatively pump flows, creating intricate structures triggers the release of a signaling molecule. The molecule that benefit the entire community (4). is initially advected by fluid flows but also increases fluid Perhaps the archetypal example of an apparently simple flows, generating a feedback loop and enabling the move- organism able to generate sophisticated behaviors is the slime ment of information throughout the organism’s body. This mold Physarum polycephalum, whose behaviors are repeatedly simple mechanism is sufficient to explain P. polycephalum’s characterized as “intelligent.” This slime mold is able to navi- emergent, complex behaviors. gate mazes by finding the shortest route between different food sources (5) and has used its ability to reconstruct the transporta- Author contributions: K.A., N.A., A.P., and M.P.B. designed research, performed research, tion maps of major cities (6). The organism can structure its con- and wrote the paper. nections to different nutrient sources to optimize its diet (7). The authors declare no conflict of interest. How the organism coordinates complex tasks in the absence This article is a PNAS Direct Submission. of a nervous system remains unknown. P. polycephalum is uni- 1To whom correspondence should be addressed. Email: [email protected]. cellular, and as it forages the slime mold develops as a retic- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. ulated network of tubes. There is no obvious organizing cen- 1073/pnas.1618114114/-/DCSupplemental.

5136–5141 | PNAS | May 16, 2017 | vol. 114 | no. 20 www.pnas.org/cgi/doi/10.1073/pnas.1618114114 Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 lme al. et Alim i.1. Fig. how explain networks. to port sufficient in seems adaptation polycephalum flow, radius P. increased tube the with to tandem response the in Moreover working (12). observation mechanism, previous organ- our matches explaining wave size, peristaltic ism the that itself implies unidenti- mechanism understand propagation remains The signal to of far us mechanism so the allows molecule of discovery signaling the the chem- fied the of a Although initiating movement. nature own is ical its molecule in enable The to increases loop travels. local feedback it causing signal- as simul- by amplitude a but generation contraction flows of fluid flow release by hijacks advected the taneously is triggers molecule The stimulus across molecule. propagation nutrient ing signal The of body: mechanism and a the experiments aggregate identify the mathematical model in a and the data build the these We in on based velocities. particles in model flow trace particles measure unrelated of to use dispersion cytoplasm We the flows. to fluid comparable the organ- cytoplasmic the velocity at across a initiating propagates with front ism amplitude amplitude contraction The site. in stimulus increase an triggers otato mltd,osre sicesdcnrs uigacnrcincce rn ficesdapiuepoaae vrtm rddse ie.(iv line). dashed (red time over propagates amplitude increased of Front cycle. in mm.) (iii contraction amplitude. 2 a contraction bar, during in of (Scale contrast changes Inset arrow). increased sudden (green mark as (C lines observed droplet vertical radii. amplitude, the Dotted contraction larger with site. of stimulation contact tubes from make through away they further preferentially where spreads (i swell colors) dataset. Tubes (hot s. field front 0 The arrow. at black in added as by Droplet points media. time same nutrient The of (B) droplet a with ulation C B A (v) (i) vrg rn rpgto pe rd ln rjcoyin trajectory along (red) speed propagation front Average i. Speed [µm/s] Amplitude frontandparticlespeeds Relative amplitudeofcontractions Network beforeandafterstimulation 20 40 60 80 0 iii rn particles front rpgtn mltd rn.( front. amplitude Propagating 5 atce detdaogtetb pera aksotdtaetre.(v trajectories. spotted dark as appear tube the along advected Particles : 4 oain fpril pe bu)adfotsed(aet)maue.(ii measures. (magenta) speed front and (blue) speed particle of Locations ) 12345 3 2 sal osleamz n ul fcettrans- efficient build and maze a solve to able is 1 A scmlxdynamics. complex polycephalum’s P. hwn otato mltd eaiet h vrg vrtels 0prosbfr tmlto.Siuainst marked site Stimulation stimulation. before periods 10 last the over average the to relative amplitude contraction showing rgtfil mgso a of images Bright-field A) i)(i)(iv) (iii) (ii)

transmitted light intensity 100 100 150 200 100 120 50 75 50 40 60 80 876 0 372 0 558 0 i ssonin shown as .polycephalum P. trace at0.35mm trace at8.4mm trace at4.3mm t(s) t(s) t(s) iii. h pedo h hnei o ymti nsaearound space in obviously symmetric amplitude in not change is the 13 instead, change site; about the stimulation of the of speed along spread a propagates The dataset with subsequently Our tubes amplitude context. in the different change entirely the an shows in done dataset dent hw la nraei silto mltd ttestimula- the at amplitude 1B oscillation (Fig. patterns in wave site the increase tion of clear fixed reduc- Analysis the a network. uniform by a shows caused A within is 1). fluid and of follows Fig. amount elsewhere in volume to tube arrow exposed in directly (green tion tubes liquid the nutritive of inflation the in over localized a extract changes spatially with and to immediate propagate begins 1 these analyzed to Meth- (Fig. and rise were and time dynamics gives Data (Materials contraction stimulus h. tube tube nutrient each 3 A along to ods). contractions 2 track of and course microscopy the bright-field using over stimulation after and before works a throughout lus Dynamics. Contraction in polycephalum P. Results ersnaiemxmlsed fprils(le,lctda pictured as located (blue), particles of speeds maximal Representative ) ,icuigapocigppte n fe stim- after and pipette) approaching including s, (−3 before network

distance along tube [mm] 0.35 8.4 4.3 mltd rn n atcesed xrce rmbright- from extracted speeds particle and front Amplitude ) otato atrsa he ifrn onsfrhrand further points different three at patterns Contraction ) time sincestimulus[s] epnst tmlswt rpgtn Change Propagating a with Stimulus a to Responds .polycephalum P. 7 5 876 558 372 0 .Atrasotdly h response the delay, short a After S1). Movie and PNAS speed front yorp ln rc in trace along Kymograph ) ;serf 2fra indepen- an for 22 ref. see S2); Movie oflo h rpgto fastimu- a of propagation the follow To | a 6 2017 16, May ewr efis bevdnet- observed first we network / Fg 1C, (Fig. µm/s | o.114 vol. particle withintube trace ofadvected i hwn hnein change showing | o 20 no. i speed particle max. and | 5137 ii).

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PHYSICS BIOPHYSICS AND COMPUTATIONAL BIOLOGY propagates more quickly along larger tubes compared with ated by the actin–myosin cytoskeleton embedded in tube walls, as tubes with smaller radii. These observations are reproducible well as the elastic restoring force FE . We represent these effects (Fig. S1). through As we tracked contraction dynamics we simultaneously ~ observed unrelated particles or vacuoles moving with shuttle flow f = (T + FE )δ(r − a)~er , [2] within tubes. These appear as dark streaks in the kymograph (Fig. 1C, ii). Because of the ephemeral nature of the particles net so the force density is localized at the tube boundary where transport velocity could not be calculated, but the streaks allow r = a(z, t), oriented radially with the unit vector ~er . Because the us to measure the maximal fluid flow velocity along a tube during thickness h of the tube wall is small compared with the base a given contraction period. Fig. 1C, iii compares maximal flow radius a0, we can approximate the elastic restoring force using speeds with speeds of the propagation front; the front propaga- the linear law tion velocity is about two- to threefold slower than the maximal E F = (a − a ), flow velocity and as such is likely to be similar to the net transport E h 0 velocity within the cytoplasm (Fig. S2). where E characterizes the wall elasticity (23). The orientation Mechanism of Signal Propagation. Experimentally we observe of the restoring force is negative for a < a0, thus counteract- a propagating change in contraction amplitude triggered by ing tension, while acting in parallel with tension for a > a0. a localized stimulus. The velocity of the amplitude front, Experiments have shown that the tension in the tube wall of 1 − 20 µm/s, is significantly slower than expected from propa- P. polycephalum oscillates with a well-defined frequency (24, 25). gation mechanisms involving elastic waves, > 0.2 m/s, or action A typical period is around 120 s. Oscillations seem independent potentials, > 2.7 mm/s, but is in the range expected from an of cytoplasmic flows because they persist if cytoplasm is replaced internally advected signal (Fig. 2). Data also rule out the diffu- by air (26), even though they become spatially uncorrelated if sive spread of a stimulus within the agar substrate: This mecha- flows are stalled (26, 27). We incorporate this into our model by nism would result in a radially symmetric increase in contraction taking T = γ cos(ωt), where the tension strength γ depends on amplitude around the stimulus site, but we observe an asymmet- the concentration of the signaling molecule c. We expect actin– ric spread along larger tubes with higher flow velocities. Exper- myosin to respond to a time average hci of the signal concen- iments suggest a basic feedback mechanism: An initial stimulus tration, so that γ = γ(hci), which for simplicity we assume to be triggers release of a signaling molecule and the molecule changes linear, γ(hci) = γhci. local wall contractions, increasing local fluid flow. Greater flow We note an important simplification of the fluid mechanics: increases the dispersion of the signaling molecule away from Because the characteristic length scale of the contraction is much its source, and the molecule continues to trigger wall contrac- larger (i.e., organism size) than the radius of the tube, the Stokes tions downstream. The process repeats itself and creates a self- equations (Eq. 1) simplify via lubrication theory (SI Text). This propagating front across the entire organism. gives the flow velocity in the tube as a response to the forcing, To test whether this mechanism can explain experiments, we now translate it into a formal mathematical model, using param- a2 ∂ u¯ = − (T + F ), [3] eters measured from the organism itself (Materials and Meth- 8µ ∂z E ods). Cytoplasmic flows are well described as a low-Reynolds- number (Re ∼ 10−3), incompressible fluid. The tube radius where u¯ is the cross-sectional average advection velocity in the a0 is much smaller than the oscillatory boundary layer thick- tube. This equation is supplemented by the incompressibility p ness ν/ω, defined by the ratio of kinematic viscosity ν and condition, an equation for conservation of mass (Eq. 5), which oscillation frequency ω. Thus, the flow velocity, ~u = (u, v), holds throughout the closed tube. with the flow longitudinal u and radial v flow components Finally, to complete our model, we must specify the dynamics in a cylindrical coordinate system, follows from the Stokes of the signaling molecule itself. This is given by equations ∂c = ∇(−~uc + κ∇c), [4] µ∇2~u = ∇p − ~f , ∇ · ~u = 0, [1] ∂t where κ is the molecular diffusivity. In the limit that the timescale where µ denotes fluid viscosity and p pressure. The force ~f repre- for diffusion across the radius of the tube (a2/κ) is fast relative to sents force on the cell wall caused by the active tension T gener- a typical time for transport along the tube, the transport dynamics

A B 15 20 action potential (with nervous system) m/s] action potential µ 15 10 (without nervous system) elastic wave 10 P. polycephalum 5 5 number of observations propagation speed [ 0 0 10-4 10-2 100 102 5070 90 110 130 150 propagation speed [m/s] tube diameter [µm]

Fig. 2. Front speed identifies mechanism. (A) Propagation speeds for elastic waves (red), action potentials (blue), (19, 20), and P. polycephalum (green). Wave velocity in a fluid filled elastic tube varies between wave speed in the elastic wall (lower bound) and the fluid only (upper bound) (21). (B) Front speed increases with tube radius as predicted by the model. Amplitude fronts along tubes of a network centrally stimulated by nutrient droplet. Data show average speed and tube diameter with one SD error measured at five locations along the corresponding route (Inset). (Scale bar, 5 mm.)

5138 | www.pnas.org/cgi/doi/10.1073/pnas.1618114114 Alim et al. Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 h bevddfuiiyo h ocnrto ace h Taylor coefficient the dispersion matches S3 A). concentration (Fig. the contraction of diffusivity of observed period The each con- ampli- over advective high flow out shuttle cancel of a tributions in region because the process, diffusive of a is propagation tude The tube. concentration the flow the signal through with both con- self-propagate thus The amplitude contraction concentration. and increased in flow and peak shuttle the a with drive peak tractions that through- tube amplitudes the signal’s contraction out self-propagating localized nonzero a in The to causes shown amplitude. rise concentration as contraction give stimulus, dynamics increased the of localized that front a find by We generated 3. Fig. signal a Dispersion. of port Taylor by (Eqs. model our Propagate Fronts Amplitude a of order the of is it as contraction. long of as period predictions model affect not does oaino h rn smaue ytepsto ftepaejm onie ihtema vrgdcnetainars h uea vr iepoint time every at tube the across concentration averaged al. by mean et marked Alim the is with front coincides Amplitude jump time. phase over the tube of the position the along by radius (C measured line). tube line). (white as mark- of (white lines front Kymograph surroundings white the (B) two and of (between versa. region location contracts vice stimulated amplitude or contraction between large expand, jump of surroundings phase region location) a When front amplitude. amplitude contraction ing and concentration signal increased of front of legend see dient, 3. Fig. amplitude increased the of diffus- speed is the frontslowsdownovertime.Inexperimentalnetworks,theslowing- advected amplitude being observations. contraction not and elevated experimental ing of with propa- region velocity agreement flow the diffusively in maximal Because the the tube, central- of than the the smaller speed within of therefore The fronts is front S3B). the gating between (Fig. in stimulus averaged ized are radius and dis- Eq. concentration (30). Taylor radius the in tube nonuniform terms to correction due persion higher-order omitted we where by given are port velocity. flow the to proportional cross- diffusion tube the augments across section diffusion quick the that Tay- shows by dispersion lor concentration averaged (Eq. cross-sectionally the dispersion of dynamics Taylor to simplify A loehrtecmlt qain oenn o n trans- and flow governing equations complete the Altogether Tube snapshots ∂ ∂ ∂ ∂ hc ktho ueadtecnetaino inln oeuewti h ueoe ie(oo gra- (color time over tube the within molecule signaling a of concentration the and tube a of Sketch (A) model. theoretical a in propagation Signal ∂ ∂ a t t c u t 2 ¯ i = = = = − − − ∂ ∂ z 8µ hc ∂ a ∂ oepoetednmc fflwadtrans- and flow of dynamics the explore to 5–8) ( z C 2 − i .Tesga nrae h mltd frda otatos(mle aisa ihcnetain n oetbihsaself-propagating a establishes so and concentration) high at radius (smaller contractions radial of amplitude the increases signal The ). c τ ∂ − ∂ a z vratimescale a over 2 uc c ¯ κ  time =16periods u ¯ , time =8periods time =4periods time =2periods + γ
+ hc u ¯ 48κ  2 i a cos(ω κ 2 + seEq. (see u ¯ 48κ t 2 + ) a 2,2) pcfigthe Specifying 29). (28, 7) 2 τ  h rcs au of value precise The . E h ,weeflwvelocity flow where 7), B ∂ ∂ (a z distance along tube c ) 0.8 0.6 0.4 0.2 − 1 0 , a 015 10 5 0 8 0 ) time-averages  is euse we First , time [periods] [8] [6] [7] [5] τ a ftetm-vrgdsga ocnrto;sra scue yflw The flow. by caused is spread concentration; signal time-averaged the of Map ) yprmtr htcnrlflwvlct codn oEq. to according concentration velocity radius overall tube flow in increase control an Specifically, that parameters by propagates front 1B, Fig. the in show- given exceptions because are radii; down different slowing observe of ing tubes to in speeds hard different with is effect down lotblne aacdfre ml uednmc of force dynamics tube restoring imply elastic forces and Balanced the balance. tension than almost bigger velocity, far flow are small forces elastic resulting Eq. and from tension jump mathemati- both phase In that observed Given versa. the rationalize vice can and we terms expand, cal zone can- must one tube other when entire the instead, An contracts synchronously; tube. expand) closed (or a contract within not volume fluid of contraction vation the in value jump of a phase by easily surroundings is its amplitude from contraction distinguished greatest Concentration. of region Mean the of simulations Calculation Allows Mass of with Conservation fronts compared amplitude tubes the larger 2). find within (Fig. tubes We quickly smaller more along tubes. amplitude propagate smaller in to and increase larger propagating a both induced and increase a center velocities stimulated front We radius. that tube prediction with model’s the tested then viscosity .polycephalum Generation. Gradient P. Contraction-Driven by Calculated Is Size System signaling mean the calculate concentration. to molecule system the enables propagation at concentration averaged the time of given mean a spatial the just is tration Fg ) otato n o yaisaeas elcpue by jump captured well phase also are the Eq. dynamics flow predicts and Contraction exactly 3). (Fig. dynamics concentration from tube’s the is, the that compared small is radius contractions base ampli- radial The the contracts, extends. of tube tube tude the the concentrations concentration smaller for reference whereas this of than jump larger phase are the where tion nsmltos efidta h rn pe ssrnl affected strongly is speed front the that find we simulations, In h c ¯ 9 (t ( )i i.S4 Fig. tube radius dt d ak h ieaeae inln oeueconcentra- molecule signaling time-averaged the marks µ R [m] a n uelength tube and 0 L a = π h .Ti nigipista h ehns fsignal of mechanism the that implies finding This ). 0 c ¯ 4.5 5 5.5 a rmcnevto ffli ouei h tube, the in volume fluid of conservation From . 10 π a (t 2 0 (z -5 .Tepaejm scue yconser- by caused is jump phase The 3B). (Fig. )i dpstewvlnt fiscnrcin to contractions its of wavelength the adapts − )dz = C PNAS distance along tube E γ R c 0.8 0.6 0.4 0.2 /h 0 = 0 0 1 0 L hc 015 10 5 0 nrae rn eoiy hra fluid whereas velocity, front increases (hc | tflosta h eeec concen- reference the that follows it , (z a 6 2017 16, May L (z , eraei.I neprmn we experiment an In it. decrease t , π time-averaged signalconcentration )idz t time [periods] h − )i cus hr concentrations Where occurs. .polycephalum P. IText). (SI and S2, Movie a c ¯ eso strength tension , | (t cos(ω )i o.114 vol. h c ¯ (t | ewr tits at network t )i i.S1A. Fig. o 20 no. ). calculated γ | and , 0 0.5 1 1.5 2 10 5139 [9] In 6. 6. -3

PHYSICS BIOPHYSICS AND COMPUTATIONAL BIOLOGY match a network’s longest dimension, independent of network nism explains the experimental observation of a self-propagating size (12). To test whether our model elucidates this adaptation increase in contraction amplitude across a body; moreover, the we probe contraction dynamics and begin with a single tube model solves the puzzle of how an individual can measure its encompassing three wavelengths of radial undulations coupled own size and match the wavelength of pulsatile contractions to a slightly randomized but otherwise constant concentration to its size. The mechanism also reveals how P. polycephalum of signaling molecule (Fig. 4). Over time, undulations in the finds the shortest route through a maze (5) or connects multi- tube’s radius and the resulting perturbations in concentration die ple food sources with shortest routes into an efficient transport out in favor of a single contraction wavelength coupled to sin- network (6, 31). gle concentration wavelength; this pattern persists through time. The signaling molecule remains unidentified but among the Although the ad hoc expected pattern would be a uniform con- molecules known to oscillate in P. polycephalum (ATP, cAMP, centration, uniformity would require homogeneous contractions H +, and Ca2+), the most likely candidate is calcium. Calcium of the tube and this would violate conservation of fluid volume. regulates actin–myosin dynamics and these dynamics drive net- Thus, the closest state to a uniform concentration, namely a sin- work oscillations (25). Contrary to electron micrographs indicat- gle wavelength, is observed, which itself entails a measure of sys- ing calcium release during contraction and sequestering during tem size and single wavelength in contractions. relaxation (32), direct measurements observe calcium at low con- centrations within contracting tubes of P. polycephalum, and at Signal Propagation Increases Flow Disproportionately Through high concentrations in relaxed tubes (33). The latter is in agree- Shorter Routes and Therefore Drives Selection of the Shortest Route. ment with the dynamics of our model’s signaling molecule. Cal- P. polycephalum is notorious for its complex morphological cium is a universal driver of metabolism (34), and it is fascinating dynamics; for example, the organism can find the shortest route to speculate that it also drives the rich and complex behavioral through a maze connecting two food sources (5). Both food dynamics of slime molds. sources are external stimuli that would, according to our obser- P. polycephalum is a charismatic model and a tool to under- vations and model, trigger self-propagating fronts of increased stand how other apparently simple organisms generate sophis- contraction amplitude. Fronts would propagate into the network ticated behaviors. The mechanism of communication we iden- from both ends of the incipient route. Increases in contraction tify involves basic features: a signaling molecule, fluid flows, and amplitude will increase the flow rate in a route proportional to an interaction between the signal and fluid flows. These features the average amplitude increase along the entire route. There- may be common to thousands of species (35, 36) and even to fore, longer routes will experience a smaller increase in flow species distantly related to slime molds, including, for example, rate compared with shorter ones. Following the reasoning of fungi. The mechanism is likely to be a general one and may serve Tero et al. (6) tube radii will grow because of an increased as a broad explanation for the complex behaviors of many organ- flow rate at the expense of tubes with less flow rate. Thus, isms without nervous systems. the self-propagating amplitude fronts strengthen the shortest route between the two food sources by increasing the flow rate Materials and Methods along it. The flow feedback-driven mechanism of signal propaga- Preparation and Imaging of P. polycephalum. Plasmodia of P. polycephalum tion explains P. polycephalum’s ability to find the shortest route (Carolina Biological Supply) were grown on 1.5% (wt/vol) agar without through a maze (5) and its ability to link multiple food sources nutrients and fed every other day with oat flakes (Quaker Oats Co.). Twenty- four hours before imaging, newly colonized oat flakes were transferred to into an efficient transport network (6, 31). a new agar surface. Each plasmodium was allowed to explore the agar sur- face and form a mature network of well-defined tubes. Sections of network Discussion were prepared by scraping away excess tissue immediately before imaging The slime mold P. polycephalum coordinates complex behaviors so that the entire network would fit within the field of view. Networks were using a simple feedback. An external stimulus triggers a change imaged using transmitted light on a Zeiss Axio Zoom V16 stereomicroscope. in contraction amplitude and the amplitude front propagates Images were taken every 3 s. A 1-µL drop of medium was used to stim- with a velocity equivalent to a particle’s diffusive transport. These ulate the network after approximately 1 h of recovery time and imaging experimental data and a model elucidate the mechanism of com- continued for 2 to 3 h. The medium used for stimulus followed the recipe munication across the network: The stimulus triggers the release of Daniel and Rusch (37) with hematin (5 µg per mL) replacing the chicken embryo extract (38). The medium gave a stronger and more reproducible of a signaling molecule advected with cytoplasmic flows. The response, compared with glucose (25) alone. Adding a source of protein, for molecule hijacks flows by increasing local contraction amplitude example tryptone or yeast extract, to glucose worked best. and generating additional cytoplasmic flows to carry itself fur- ther into the network, where it again increases local contraction Image Analysis. Key time-variant parameters were calculated from every amplitude and again generates cytoplasmic flows. This mecha- point of a network using custom MATLAB (The MathWorks) code. Briefly,

A tube radius 10 -5 [m] B signal concentration 10 -4 0 5.004 0 1.67

1.668 5.00395 1.666 0.5 5.0039 0.5 1.664 5.00385 1.662 distance along tube distance along tube 1 5.0038 1 1.66 0 10203040 010203040 time [periods] time [periods]

Fig. 4. Decay into a stationary state of a single wavelength matching organism size. The initial condition is set as three undulations in radius along a tube (A) and a slightly randomized but otherwise constant concentration of the signaling molecule concentration (B). The feedback between concentration and contraction amplitude drives the system to a single undulation in radius and concentration; the wavelength matches the tube length. Note that high and low concentrations and radii at early time points are not representing their true value to accommodate the range generated at final time points.

5140 | www.pnas.org/cgi/doi/10.1073/pnas.1618114114 Alim et al. Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 1 ulrC,Fh J(92 hrceitc fwv-rpgto n energy-distribu- and wave-propagation of Characteristics (1982) FJ Fahy CR, Fuller 21. 0 esS,Mci O ec W(99 mus odcini sponge. a in conduction Impulse (1999) RW Meech GO, Mackie SP, Leys 20. Taylor increase to Pruning (2016) MP Brenner A, Pringle N, Andrew K, Alim S, Marbach 18. 9 ulc H ordeG (1965) GA Horridge TH, Bullock 19. motility. and structure Plasmodial (1982) D Kessler 17. concen- cGMP and cAMP in Changes (1988) Y Kobatake T, Nakagaki Y, Mori T, Energy Ueda (1983) 16. KE Wohlfarth-Bottermann Z, Baranowski Z, Shraideh W, Korohoda 15. and processes Auto-oscillatory by (1997) W propagation Alt DA, Information Pavlov YM, (1993) Romanovsky VA, H Teplov Shimizu 14. M, Yano Y, Miyake network K, Random Natsume (2013) A 13. Pringle MP, Brenner F, Peaudecerf G, Amselem polycephalum. K, Physarum Alim in chemotaxis 12. of Control (1976) EB Ridgway AC, Durham 11. of strands protoplasmic in contractions Oscillating (1977) KE Wohlfarth-Bottermann 10. lme al. et Alim frequency model a to Param- chosen MathWorks). (The eters MATLAB using scheme Crank–Nicolson weighted Simulation. same the along speeds particle propagation. maximal of typical, routes measure to Kymographs used site. applica- also stimulation were the the with kymographs intersecting before tubes fronts, along propagation immediately created measure were periods and visualize To 10 stimulus. a the of tion in tube. normalized windowed were documented a a amplitudes using values changes, within then to relative and fit record To oscillations point entirely transform. the Hilbert every still detrending at first would was by radius amplitude that Contraction recovered tube extracted. were circle the contractions of largest amplitudes recover Local the to and used finding nodes was of by plasmodial map skeleton the connected just a The extract into edges. separated to and thresholded skeletonized was then series tubes time a from image an .Nkgk ,Ymd ,T H, Yamada T, Nakagaki 5. .Myk ,Td ,Yn ,SiiuH(94 eainhpbtenitaellrperiod intracellular between Relationship (1994) H Shimizu M, Yano H, Tada Y, Miyake of 9. plasmodia in Chemotaxis (1982) Y Kobatake com- solves T, organism Ueda Amoeboid (2010) 8. S Simpson M, Beekman T, design. Latty network A, adaptive Dussutour inspired biologically 7. for Rules (2010) al. et A, Tero 6. in channels by facilitated transport Liquid (2013) al. Dispersal et systems: JN, cord Saprotrophic Wilking (2009) MD 4. Fricker DP, Bebber J, Hynes L, bacteria. Boddy in Chemotaxis 3. (1975) HC Berg 2. (1980) JT Bonner 1. 0 = in nclnrcleatcsel le ihfluid. with filled shells elastic cylindrical in tions 202:1139–1150. Invertebrates in dispersion (Academic, JW 145–210. Daniel pp HC, 1, Aldrich Vol eds York), Cycle, New Cell and Nucleus, Organisms, Didymium: htaodnei lsoi fa liosri fPyau polycephalum. Physarum of strain light albino blue Photobiol an and tochem of UV plasmodia accompanying in activity contractile photoavoidance and fibrils birefringent tration, polycephalum. Physarum in activity Res contraction Tissue Cell oscillatory of regulation metabolic eds Motion, in mechanisms feedback cephalum Ca of change pattern spatio-temporal USA Sci Acad Natl in peristalsis Biol Cell periodicity rhythms, dependence. radial temperature and and analysis longitudinal of tensiometry Simultaneous Physarum: in change Funct eds environment external Cycle, and Cell modulation 112–144. pp and 1, Nucleus, Vol York), Organisms, New (Academic, Didymium: JW Daniel and HC, Physarum Aldrich of Biology Cell challenges. nutritional plex 327:439–442. 407:470. biofilms. time. and space in mechanisms Princeton). 50 19:363–370. ,wl height wall µm, 69:218–223. rcNt cdSiUSA Sci Acad Natl Proc ω ihrpliestimulation. repulsive with oe Eqs. Model 0 l ,DushA unG Srne,Bsl,p 83–92. pp Basel), (Springer, GA Dunn A, Deutsch W, Alt = hsrmpolycephalum Physarum hsrmpolycephalum Physarum Femn a Francisco). San (Freeman, 231:675–691. 2s yai icst fcytoplasm of viscosity dynamic 120s, 110:13306–13311. 47:271–275. h vlto fCluei Animals in Culture of Evolution The t 20)Mz-ovn ya meodorganism. amoeboid an by Maze-solving (2000) A oth ´ hsrmplasmodium Physarum .polycephalum P. 5–8 h rcNt cdSiUSA Sci Acad Natl Proc = Mycoscience eesle ueial sn custom, a using numerically solved were 110:848–852. 0.1a tutr n ucini h evu ytm of Systems Nervous the in Function and Structure 0 elSrc Funct Struct Cell raie udflw cosa individual. an across flows fluid organizes networks. 2 uelength tube , x Biol Exp J + n e ipy Bioeng Biophys Rev Ann ocnrto throughout concentration 50:9–19. nld aetb aisof radius tube base a include on Vib Sound J motility. hsRvLett Rev Phys 67:49–59. hsrmplasmodium. Physarum 107:4607–4611. elBooyo hsrmand Physarum of Biology Cell 18:111–115. L yaiso eladTissue and Cell of Dynamics = hsrmpolycephalum. Physarum µ 81:501–518. m contraction cm, 0.5 PictnUi Press, Univ (Princeton = 117:178103–178105. 4:119–136. 6.4 · hsrmpoly- Physarum ailssubtilis Bacillus 10 −3 elStruct Cell x Biol Exp J s m Ns/ Science Nature Pho- Proc 2 J 2 cmd G imn ,Scmn 19)Serfil apn natnntok by networks actin in mapping field Shear (1996) E Sackmann F, Ziemann the FG, of Schmidt position the 42. destabilizes contractility actomyosin Polar (2011) al. et J, Sedzinski 41. change cells dividing how Understanding (2012) A Surcel T, Luo and YS, Kee DN, recovery Robinson Photobleaching 40. (1997) AS Verkman CP, Hoang myxomycete. R, plasmodial Hematin-requiring Swaminathan (1962) 39. HP Rusch J, Kelley JW, Daniel 38. 7 ailJ,RshH 16)Tepr utr of culture pure The (1961) HP Rusch JW, Daniel 37. (2007) eds M, Michalak J, in Krebs contractility 34. to efflux Ca2+ calcium of oscillations Simultaneous cytoplasmic (1981) N of Kamiya F, Relation Matsumura Y, Yoshimoto (1982) 33. H Kuroda R, Kuroda (2012) G 32. Varma pipes. in K, dispersion Mehlhorn shear V, of Bonifaci model complete tube. 31. A a through (1994) flowing AJ fluid Roberts a GN, in Mercer solute a 30. of dispersion the On (1956) tube. R a Aris through slowly 29. flowing solvent in matter soluble of Dispersion (1953) G Taylor 28. 6 akrL,Sehno L(06 h pce rbe nmxmctsrevisited. myxomycetes in problem species The (2016) SL Stephenson LM, Walker 36. (1994) H Stempen SL, Stephenson 35. transmis- signal and Synchronization (1981) K Wohlfarth-Bottermann U, Achenbach of 27. relation Phase (1984) K Wohlfarth-Bottermann Z, Hejnowicz I, Hinz behavior K, cell Samans the 26. in formation pattern and oscillations Intracellular (1993) T Ueda 25. Delano tube. elastic JP, an Rieu in fluid 24. viscous of pumping Peristaltic (2011) NJ Balmforth D, Takagi gra- 23. by induced waves Propagated (1980) KE Wohlfarth-Bottermann Z, Hejnowicz 22. o ysnatvt nHL el 4)t estimate c to (41) cells of HeLa stress ratio in elastic activity myosin-generated myosin of for value calculated the T used we modulus 3) n oeua ifsvt fasalrmlcl ie,APi cytoplasm) in ATP (i.e., molecule smaller a κ of diffusivity molecular and (39), eso teghwscoe ob ntelna eieo h rn velocity front the of regime linear periods the in two be γ about to chosen as was chosen strength tension was window time averaging e aufrce eplia(...MPB sa netgtro h Simons the of Investigator an is M.P.B. Foundation. Akademie (K.A.). Deutsche Leopoldina the Division Naturforscher and and der DMS-1411694, DMR1420570 Grant Grant Sciences Mathematical Center of Engineering Materials and Harvard Science through Foundation Research Grant Science through National Program Science the Frontiers RGP0053/2012, Human the by supported was work ACKNOWLEDGMENTS. = = = = sn antctweezers. magnetic using furrow. cytokinetic 48–73. pp 7, shape. rotational and translational Cyto- protein cells: fluorescent and green solution diffusion. by in probed GFP-S65T protein viscosity fluorescent plasmic green of decay anisotropy Bacteriol endslbemedium. soluble defined of strand plasmodial 1:433–443. permealized the in generation tension and polycephalum. Physarum Biol Theor Math Appl Ind A Soc A Soc R Proc tist OR). Portland, Ed. 1st York), Physarum. of strands protoplasmic in sion stages. plasmodial different to 241–250. application its and device in cycles contraction oscillatory 167–181. of of amoeba 20150099. large migrating in Mech Fluid of plasmodia within 150:144–152. factors physiological of dients p 0P during Pa 50 10 10 Physarum. E 167:319–338. − − /ρ(1 235:67–77. 0m 10 ν d gla H it Esve,NwYr) Vol York), New (Elsevier, D Wirtz EH, Egelman eds Biophysics, Comprehensive 0Pa. 50 = 84:1104–1110. ipy J Biophys − 309:121–133. 4) eoiyo neatcwv ntb alflosusing follows wall tube in wave elastic an of Velocity (42). 0.4 672:196–218. 2 219:186–203. s eutn in resulting /s, ν dRnigL(ekr e ok,pp York), New (Dekker, L Rensing ed Morphogenesis, and Oscillations 2 11:499–521. -yr ,Tkg ,Tnk ,Nkgk 21)Proi traction Periodic (2015) T Nakagaki Y, Tanaka S, Takagi H, e-Ayari ¨ .Frafli lsi aewtrsprmtr r sd The used. are parameters water’s wave elastic fluid a For ). itotlu discoideum Dictyostelium Nature 72:1900–1907. etakMr rce o rifldsusos This discussions. fruitful for Fricker Mark thank We PNAS elSci Cell J e Microbiol gen J u ipy J Biophys Eur 476:462–466. Re Physarum | hsrmpolycephalum. Physarum acu:AMte fLf rDeath or Life of Matter A Calcium: yoyee:AHnbo fSieMolds Slime of Handbook A Myxomycetes: = 53:37–48. a 6 2017 16, May 2ua 24:348–353. 0 25:47–59. Planta lsoi:I eilifae registration infrared serial A I. plasmodia: /ν Physarum ∼ hsrmpolycephalum Physarum 08 oetmt h elastic the estimate To 0.0008. 151:584–594. laae(0 and (40) cleavage | hsrmpolycephalum. Physarum a opt hretpaths. shortest compute can o.114 vol. E = nedsi yl Res Cycle Interdiscip J 4P n Poisson’s a and Pa 44 o o Interface Soc Roy J Physarum. | τ o 20 no. = Esve,New (Elsevier, napartially a on T 9 ,and s, 190 /E elMotil Cell (Timber, | = Planta rcR Proc 5141 p J Jpn 1.15 Pro- 12: 15: J J J

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