Supplementary Material for Wave-pinning and cell polarity from a bistable reaction-diffusion system

Transcription

1 Supplementry Mteril for Wve-pinning nd cell polrity from bistble rection-diffusion system Yoichiro Mori, Alendr Jilkine nd Leh Edelstein-Keshet Model Comprisons We concentrte here the three systems to compre: Wve-pinning (WP b = b (k + γ = b K + (k + γ K + δ + D, + δ + D b b. ( (b Locl ecittion globl inhibition (LEGI A I = k S(t, k A, ( = D I + k IS(t, k I I, (b = k A( T k I. (c Activtor depleted substrte model (ASDM b ( s = b + s ( s = b b b + s r + D, r b b + D b b. ( (b. The LEGI model In Eqs., A is the ctivtor, I is the inhibitor nd is the response element, nd S(t, is the eternl stimulus (,. For time-invrint stimulus such s S(t, = (c + c, (unique stedy stte vlue cn be computed (. The response to constnt, sptilly uniform S = c (i.e. for c = is independent of S. To generte Fig, we first set S = c to obtin the stedy stte levels, nd then used trnsient stimuli described in the Appendi of our pper (e.g., S = c + c, together with prmeter vlues k = k =, k = k =, k I = k I =, T =, D =. To represent the removl of stimulus we reset the vlue of S to c. In Figure, we demonstrte prototype responses of the LEGI model to the sme repertoire of stimuli used in our wve-pinning tests. As seen in Fig. (, trnsient stimulus of the left edge of cell, ks loc. cuses trnsient response in t t =, but tht response decys rpidly fter the signl is turned off.

2 (Note return to homogenous distribution of by t =. The sme phenomenon is observed in Fig. (b in response to trnsient grded stimulus, k grd S. A direct result of the LEGI s pssive redout property, model cell rpidly shifts polrity in response to grdient reversl, demonstrted in Fig. (c. However, sptilly noisy stimulus ( perturbed rest stte fils to produce spontneous polriztion, regrdless of its mplitude. This is illustrted in Fig. (d, nd stems from erlier comments bout the bsence of pttern-forming mechnism in the LEGI model. Eqs. hve unique sptilly homogeneous stedy stte A s, I s, s when S = c. As shown in Fig., the LEGI model hs n incresing response to stronger grdients, without threshold. When stimulus is first turned on, the initil time behviour before inhibition kicks in is pproimted by γ k A s (units of seconds.. The ASDM system For Eqs., typicl prmeter vlues, s =.4, s =., b =, r =.4, b b =.6, r b =, D u =., D v = were kindly suggested by Prof. H. Meinhrdt (personl communiction, who lso noted tht r b cn be set to zero with no loss of functionlity. With these prmeters, the sptilly uniform stedy stte of the system, stble spirl, occurs t u =.5, v = It is esy to check tht this set of prmeters stisfy the conditions for Turing pttern formtion on periodic domin s ssumed in most simultions of cell polrity by Meinhrdt. To choose set of prmeters for this system tht would permit resonble comprison of behviours to be mde to the WP model, we scled the system so tht rtes of diffusion mtch those of the ctive nd inctive ho proteins, s in our wvepinning model (since these prmeters cn be resonbly determined from known biology. Scling time by fctor of 5 leds to the new set of prmeters s =., r =., r b = (units s, b b = (conc/s, s =. (conc, D u =.5, D v = (units of µm /s. The prmeter s is not chnged by this rescling, s it hs units of [concentrtion]. This set of prmeters is idelly suited to periodic domin, s it leds to unstble wvenumbers of the form q = nπ/l for n =,,, implying tht not only the polr pttern, but others with up to peks could occur. By djusting the slow diffusion to D u =.µm /s, while retining ll other vlues, we found tht the system ws even better suited to no-flu D domin. It hs identicl rtes of diffusion s our wvepinning model nd is thus suitble for numericl comprisons. Our finl choice for prmeters ws consequently s =., r =., r b =, b b =, s =., D u =., D v =. The requirement tht polr pttern cn be ecited (wvenumber of the form q = nπ/l for n = on D domin of length L = µm with no-flu boundries imposes strict constrints on rte constnts. The resulting system ws consistent with polr ptterns on this D domin, but the mode n = ws lso Turing unstble, nd so occsionlly higher mode (with two peks, one t ech edge would lso pper. Observe tht rtes s, r re one order of mgnitude slower thn the corresponding rtes γ, δ in the wve-pinning model. This choice ws not rbitrry. ther, it ws requirement for Turing instbility of Eqs. s eplined in Section. of our pper. Stimuli were pplied in two wys, either by dding ks loc, kgrd S, etc. to the term in lrge brces in Eqs., or by setting s = S(t, = ks loc, kgrd S. Similr results were obtined for both methods. Supplementry Figure illustrtes the distinct temporl nd sptil behviours of the ASDM mechnism tested with the sme repertoire of stimuli in these wys. Pnels (,b show tht polrity evolves in response to both loclized nd grded trnsient stimuli, ks loc, kgrd S. Qulittive fetures of the response re distinct from those of wvepinning (Fig,b in the pper: first, in the substrte-depletion model, the reltive chemicl levels t the two ends of the cell ( = vs = L chnge grdully, tking mny tens of seconds to crete n pprecible mcroscopic polriztion. By t = 4s, the polrity is still wek. By t = s, the polrity response begins to pproch its miml mplitude. By comprison, in wvepinning, very fst initil response cretes strong chemicl difference between the cell ends (by t = or erlier nd lter the propgting ctivtion front moves further into the cell. Second, comprisons of pnels (c in Supplementry Figure nd Fig. in the pper shows tht polrity generted by the ASDM system tends to freeze once creted, nd fils to respond by reversing when the stimulus grdient is reversed, unlike the wve-pinning model. (Genericlly, third intermedite with longer time-constnt is ssumed to inhibit the ctivtor nd unfreeze the system in chemotctic models, e.g. (. Finlly, comprison of the sensitivity of the two models to sptilly noisy initil conditions (Pnels d in the two figures shows tht ASDM is highly sensitive to signls of rbitrrily

3 low mplitude (relted to its Turing instbility. By comprison, response in the wvepinning model is triggered only by stimuli bove some threshold. In response to rndom initil conditions, both models occsionlly produce non-polr pttern, with either more thn one pek (ASDM, or pek in the center of the domin (WP tht is metstble stte ( (b t= (c (d Figure : The LEGI model (Eqs., showing responses (solid lines to the sme four types of stimuli nd/or initil conditions (dotted lines s for the wve-pinning model. In ech cse, strting t t = with the stedy stte, trnsient stimulus ks loc, pplied t the left cell edge in (, or s grdient, kgrd S cross the cell in (b leds to trnsient response t t = tht disppers by t =, fter the stimulus is turned off. eversing the grdient leds to reversl of polriztion (c. ndom initil conditions (d do not led to ny spontneous polriztion, regrdless of noise mplitude. Sensitivity nd persistence In Fig., we compre the responses of ll three models to both persistent (left nd trnsient (right stimuli of vrious grdient strengths. Severl differences re noteworthy. The WP model (top requires grdients lrger thn some threshold to respond, unlike both ASDM nd LEGI. The LEGI model (middle responds

4 .5.5 t=8 t=4.5.5 t=6 t=8 t= ( (b t=4 t=4.5.5 t=6.5.5 t= (c (d Figure : The ASDM system (Eqs. showing responses to the sme four types of stimuli used in the wve-pinning model nd in Supplementry Fig.. Sptil profiles for the ctivtor (solid lines t vrious times (t =, 4,..., 4s re shown here to emphsize nd compre the time scle tken for mcroscopic polriztion to be evident. In (, it tkes well over 4s for the stimulus t the left cell edge to crete noticeble chemicl difference between front nd bck of the cell, nd the mgnitude of polriztion creeps to its full mplitude only by bout t = s. A similr outcome occurs for sptil grdient (b. (Compre timescle with plots on Fig.,b in the tet. The polriztion persists even when the stimulus is removed (in both ( nd (b. Unlike both wve-pinning nd LEGI models, once formed, the polrized pttern is frozen nd does not respond by repolrizing when the grdient is reversed s shown in (c. The model dmits Turing D instbility nd is sensitive to (rbitrrily smll perturbtions of the homogenous stedy stte, unlike the previous two models. 4

5 in wy tht increses with stimulus strength, but forgets the stimulus once it is turned off. eferences. Levchenko, A., nd P. Iglesis,. Models of eukryotic grdient sensing: Appliction to chemotis of moebe nd neutrophils. Biophys. J. 8:5 6.. Kutscher, B., P. Devreotes, nd P. Iglesis, 4. Locl Ecittion, Globl Inhibition Mechnism for Grdient Sensing: An Interctive Applet. Science s STKE 9:pI.. Meinhrdt, H., 999. Orienttion of chemotctic cells nd growth cones: models nd mechnisms. J. Cell Sci. :

6 ( (b (c (d (e (f Figure : Comprison of sensitivity nd persistence of the wve-pinning model (WP (top, the LEGI model (center nd the substrte-depletion model (ASDM (bottom in response to uniform nd successively stronger grded stimuli. In ech pnel we show the profile (of the ctive form in Eqs., of the response element in Eqs., nd of the ctivtor in Eqs. t t = in response to the stimuli S( =.(+m where m =,,, 9. Left pnels: grdient on for entire durtion; ight pnels: trnsient grdient, turned off t t = 5. The ASDM (e,f is most sensitive: it responds to ll but the uniform stimulus by polrizing. The wve-pinning system (,b ehibits threshold, nd LEGI produces response tht increses with signl. Both WP (b nd ASDM (f but not LEGI (d persist fter the stimulus is removed. 6