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(Press F5 to see the movies). Some general remarks how the computer simulations are performed.

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  1. (Press F5 to see the movies)

  2. Some general remarks how the computer simulations are performed In a simple activator-inhibitor mechanism the activator a has a non-linear positive feedback on its own production and on that of the inhibitor b. The following equation describes the concentration changes per time unit for the activator and the inhibitor: sis the competence or source density, describing the ability of the cells to perform the pattern-forming reaction; raand rb are the removal rates, ba and bb are small activator-independent production rates. For instance, ba is required to initiate activation during regeneration if all activator-producing cells are removed. To perform computer simulations, such reactions have to be rewritten as difference equations. The new activator concentration in the cell i is given by the existing concentration plus the change after a short time interval: ai,t+1 = ai,t + si (ai,t2 + ba)/bi,t – ra ai,t + Da( (ai-1,t - ai,t ) + (ai+1,t – ai,t) ) The last term describes the loss or gain by diffusion, which depends on the concentration difference between a particular cell and its neighbors. With many such iterations, the total time course of the activator and inhibitor concentrations can be calculated.

  3. The sequential updating The new concentrations have to be calculated for each cell. The updating has to proceed sequentially in a loop, i is the running index, going from 1 to n (number of cells). axt(i+1) axt(i) axt(i+1) 1 i n The concentration are stored in arrays. For instance, an array axt(1, i) may contain the activator concentration of the cell i, axt(2,i) that if the inhibitor. For the calculation of the concentration change in a particular cell, first the array values (e.g., axt(1,i) for the activator) are stored as local concentrations (e.g., a or b).

  4. Boundary conditions Special conditions have to be introduced for the cells at the boundaries as they do not have left or right neighbors. Impermeable boundaries can be introduced by assuming virtual cells to the left and to the right of the terminal cells that have the same concentrations as the terminal cells. The concentrations of the cell left to the actual cells are denoted as al., bl,…: Boundary condition: impermeable al = ax(1) ax(n+1) = ax( n) n 1 (no exchange by diffusion takes place if two cells have the same concentrations, i.e., nothing leaks out at the boundaries)

  5. The code for a BASIC-program First calculated are the new concentrations that result from decay and diffusion. This calculation is made separately since it is the same for all interactions. The result is stored, e.g., for the activator, in olddecaydiffuseA. Next calculated is the concentration change due to the reaction proper. In a program written in BASIC the corresponding program code looks as follows: al = axt(1, 1): bl = axt(2, 1) ‘ left virtual cell; boundary condition: impermeable FOR i = 1 TO n' i = current cell, n = right-most cell a = axt(1, i) 'local activator-concentration b = axt(2, i) 'local inhibitor-concentration s = axt(0, i) 'local ability to perform the pattern-forming reaction ‘REM olddecaydiffA: = old a-concentration minus decay pluschange due to diffusion: olddecaydiffA = a - RA*a + DA * ((al - a) + (axt(1, i + 1) - a)) olddecaydiffB = b - RB*b + DB * ((bl - b) + (axt(2, i + 1) - b)) ‘ adding the concentration change by the reaction und updating of the array: axt(1, i) = olddecaydiffA + s * (a * a / b + ba)’ updating the activator concentration axt(2, i) = olddecaydiffB + s * a * a ’ updating the inhibitor concentration al = a: bl = b ‘Important: the original, not the updated concentrations (!) have to be used as left-cell concentrations when the subsequent cells are calculated. NEXT i

  6. Equation and computer code Thus, the concentration change per time unit as used in the original equation… …and the code for the updating the concentrations in the program look very similar. Both are easy to read: ax(i) = olddecaydiffA + s *(a * a + ba ) / b bx(i) = olddecaydiffB + s * a * a In the following, only these crucial lines of the program code are given. They provide all information needed to repeat a simulation Links to obtain the compiler, a working environment and program examples are provided at the end of this PPT.

  7. Formation of a polar pattern by an activator-inhibitor reaction in a growing field (Fig. 1) FOR i = ja TO js: GOSUB olddecay ‘olddecaydiffA etc. is calculated there axt(1, i) = olddecaydiffA + s * (A * A / B + ba) ‘ja is the left-most cell axt(2, i) = olddecaydiffB + rb * A * A + bb ‘js is the right-most cell NEXT I ‘(changing during growth) Parameters used KT= 65:KP= 400:KX= 16:KY= 18:KD= 1:KI= 1:KE= 1:KR= 3:KN= 2:KG= 4: K1= 1:K2= 2:K3= 0:K4= 0: DA= 0.0150:RA= 0.0020:BA= 0.0300:SA= 0.0000:CA= 0.0000:AA= 1.0000:GA= 1.0000: DB= 0.4000:RB= 0.0030:BB= 0.0000:SB= 0.0000:CB= 0.0000:AB= 1.0000:GB= 1.0000: __________________________________________________________________________ DA, DB..: Diffusion constants, RA, RB: removal in each iteration, BA, BB: baseline (activator-independent) production rates, AA, AB: possible special concentration in the initial concentrations; GA, GB: global initial concentrations of A, B…; other parameter for coupling of the reactions, SA is usually used for saturation of the autocatalysis. Parameters starting with H….N are always integer parameters and used for program control: KT = Number of displays; KP = number of iterations between displays ( e.g., lines in the plot at right); total number of iterations = KT * KP; KX = left-most cell, KY = right-most cell (or cells in y-direction in two-dimensional simulations; KD = type of display, KI selects initial conditions; KE selects the reaction type; KR the degree of random fluctuations in %; The production term s in cell i is set equal to the removal plusminus fluctuations (s = axt(0, i) = RA * (1 + KR/ 100 * (RND - .5)); RND is a random number 0<RND<1. KG number of displays after which growth occurs (in this case = 4 as visible in the plot)

  8. In a growing field, pattern formation starts whenever a certain extension is exceeded (Fig. 1) The result is a graded concentration profile. In other word, in growing fields, even if initiated by random fluctuations, a reproducible polar pattern results.

  9. Pattern regeneration in fragments (Fig. 1) In a non-activated fragment, the inhibitor concentration declines until a new activation is triggered. If some activator-producing cells are included in the non-activated fragment (blue arrow), the non-activated fragment will regenerate with the original polarity. In the fragments the final maximum activator concentration will be lower since there is less space into which the inhibitor can escape.

  10. Polarity reversal in a non-activated fragment If no activator-producing cells are included in the non-activated fragment, the non-activated fragment can regenerate with polarity reversal (as observed in regenerating sea urchin embryos and in stolons (see Fig. 2c). The reason is that after fragmentation the initially non-activated terminal side has the lowest inhibitor concentration (blue arrow). In other systems such as hydra, the system generates an intrinsic asymmetry such that regeneration will always occur with the original polarity (see Fig. 6 and 7)

  11. Simulations of experiments with stolons: trigger of branches (Fig.2 D-F) FOR i = ja TO js: GOSUB olddecay ‘ a = activator : b = inhibitor axt(1, i) = olddecaydiffA + s * (a * a / B + ba) axt(2, i) = olddecaydiffB + rb * a * a + bb - sb * c * B NEXT I ‘(c makes increased inhibitor destruction ‘ = mechanical stimulation ‘(in the initial conditions c is set to zero everywhere except ‘the two special cell (c = 1; black arrows in D - F) KT= 40:KP= 80:KX= 1:KY= 36:KD= 1:KI= 22:KE=102:KR= 1:KN= 4:KG= 0: K1= 1:K2= 5:K3= 0:K4= 0: DA= 0.0050:RA= 0.0050:BA= 0.1000:SA= 0.0000:CA= 0.0000:AA= 0.0000:GA= 0.0000: DB= 0.4000:RB= 0.0080:BB= 0.0020:SB= 0.1000:CB= 0.0000:AB= 0.0000:GB= 0.1000: An activator-inhibitor system is used with some basic inhibitor production (bb) such that a spontaneous trigger is impossible. The mechanical stress is assumed to cause some inhibitor leakage ( -sb * c * b, c is locally set to one for two cells in the initial conditions, indicated by two black arrows in the front).

  12. Fig. 3: Simulations from the original Kybernetik-paper (1972) FOR i = ja TO js: GOSUB olddecay axt(1, i) = olddecaydiffA + s * (A * A / B + ba) axt(2, i) = olddecaydiffB + .5 * s * A * A NEXT i KT= 50:KP= 200:KX= 1:KY= 40:KD= 1:KI=382:KE=101:KR= 1:KN= 2:KG= 0: K1= 2:K2= 2:K3= 0:K4= 0: DA= 0.0030:RA= 0.0035:BA= 0.0007:SA= 0.0000:CA= 0.0000:AA= 1.0000:GA= 3.0000: DB= 0.4000:RB= 0.0045:BB= 0.0000:SB= 0.0000:CB= 0.0000:AB= 1.4000:GB=100.0000: Green: activator (a), red: long-ranging inhibitor (b), blue = competence (s) The distribution as formed above was used as the initial situation for the simulation of transplantation and graft experiments. In these simulations, the source density s (or competence - the term used in the paper) remained unchanged (in contrast to, e.g. the situation in Fig. 6 and 7) and is fixed in the initial condition as plotted. The particular distribution was chosen to be compatible with the pronounced apical dominance of the hypostome and the more weekly graded distribution expected in the remaining body column based on transplantation experiments.

  13. Simulation of ß-catenin and Wnt3a during pattern formation (Fig. 4)….. The delayed activation of sharp Wnt 3a peak (green) in relation to the smoother ß-catenin distribution (blue) is reproduced (compare with Hobmayer et al., 2000).

  14. … and ß-catenin and Wnt during regeneration (Fig. 4) During regeneration the delayed activation of the sharp Wnt 3a peak (green) in relation to the smoother ß-catenin distribution (blue) is clearly visible (compare with Hobmayer et al., 2000).

  15. Different dynamics of ß-catenin and Wnt-3 expression in aggregates (Fig. 5) The ß-catenin expression (blue) forms more cloudy patterns that sharpens in the course of time, Wnt3 and Brachyury appear directly as sharp peaks. Assumed are two positive loops. One involves Wnt-3 transcription  secretion of the signal that diffuses moderately  stabilization of ß-catenin (blue)  Wnt-3 transcription. The second involves Wnt-3  Brachyury (red)  Wnt3. The second loop can be only triggered only if the first loop has achieved a certain level. Since e.g., Brachyury does not diffuse, the peaks are very sharp. The long-ranging inhibition is assumed to result from the Wnt-3 molecules that are associated with lipid particles that diffuse much faster. red: Wnt-3 -Bra loop, blue: Wnt-3 - ß-catenin-loop; (use F5 for full display to see the movie, parameters on the next page).

  16. Different dynamics of ß-catenin and Wnt-3 expression (Fig. 4 and 5) FOR iy = kay TO jy: GOSUB roundleftb FOR ix = 1 TO js: GOSUB roundafcstore IF iex(ix, iy) > 0 THEN ‘only these cells participate in the reaction, used to simulate rounded fields axy(1, ix, iy) = olddecaydiffA + s * (cb * c * c + sa * e * e + ba) ‘ Wnt transcription axy(2, ix, iy) = olddecaydiffB + rb * a 'Wnt-K ‘ secreted Wnt axy(3, ix, iy) = olddecaydiffC + bc - cc * c * d / (sc + b) ‘ß-catenin axy(4, ix, iy) = olddecaydiffD + rd * b ‘ longer-ranging Wnt axy(5, ix, iy) = olddecaydiffE + re * a / (be + d * d) / (1 + se * e * e)' Bra needs Wnt-T, inhibited by Wnt-long (d) END IF NEXT: NEXT KT= 12:KP=1000:KX= 24:KY= 24:KD= 11:KI= 1:KE=921:KR= 1:KN= 5:KG= 0: K1= 2:K2= 3:K3= 0:K4= 0: DA= 0.0000:RA= 0.0080:BA= 0.0500:SA= 1.0000:CA= 0.0000:AA= 0.1000:GA= 0.1000: DB= 0.0050:RB= 0.0060:BB= 0.0000:SB= 0.0000:CB= 1.0000:AB= -0.0020:GB= 1.0000: DC= 0.0000:RC= 0.0003:BC= 0.0020:SC= 0.0000:CC= 0.0050:AC= 0.1000:GC= 0.1000: DD= 0.2000:RD= 0.0050:BD= 0.0000:SD= 5.0000:CD= 0.0000:AD= 0.2000:GD= 0.2000: DE= 0.0000:RE= 0.0020:BE= 1.0000:SE= 0.0000:CE= 0.5000:AE= 0.0000:GE= 0.0000: In this equation, a:= Wnt transcription; b:=secreted Wnt; c = ß-catenin accumulation in the nucleus (local, DC = 0; blue in the plot). The half life of ß-catenin increases with increasing Wnt signaling. The effect is counteracted by the more diffusible Wnt (-> d). The cell-local loop may work via Brachyury (=> e ; red) that is assumed to have a feedback on Wnt-transcription.

  17. The role of a graded competence (Fig. 6) 'Hy - ß-cat stabilization with short-ranging Wnt, feedback on competence (f) ‘ via long-ranging Wnt (d) ' competence F has influence on ß-cat stabilization (c). FOR i = ja TO js: GOSUB olddecay axt(1, i) = olddecaydiffA + s * (cb * c * c + sa * e * e + ba) 'Wnt transcription axt(2, i) = olddecaydiffB + rb * a 'Wnt-secreted, slowly diffusible' axt(3, i) = olddecaydiffC + f * bc - cc * c * d * d / (sc + b) 'ß-cat axt(4, i) = olddecaydiffD + rd * b ' WNT-L 'Wnt long range' axt(5, i) = olddecaydiffE + re * a / (be + d*d) / (1 + se * e * e)' Bra needs Wnt-T, inh by Wnt-L axt(6, i) = olddecaydiffF + rf * d ‘competence, controlled by the long-ranging Wnt (d) NEXT KT= 40:KP=2000:KX= 1:KY= 8:KD= 19:KI= 38:KE=211:KR= 1:KN= 6:KG= 2: K1= 2:K2= 3:K3= 0:K4= 0: DA= 0.0000:RA= 0.0080:BA= 0.0300:SA= 1.0000:CA= 0.0000:AA= 0.1000:GA= 0.1000: DB= 0.0100:RB= 0.0060:BB= 0.0000:SB= -0.0050:CB= 1.0000:AB= 0.0000:GB= 1.0000: DC= 0.0000:RC= 0.0003:BC= 0.0020:SC= 0.0000:CC= 0.0050:AC= -0.0050:GC= 0.5000: DD= 0.4000:RD= 0.0050:BD= 0.0000:SD= 5.0000:CD= 0.0000:AD= 0.2000:GD= 0.2000: DE= 0.0000:RE= 0.0020:BE= 1.0000:SE= 0.0000:CE= 0.5000:AE= 0.0000:GE= 0.0000: DF= 0.0000:RF= 0.0001:BF= 0.0000:SF= 0.0000:CF= 0.0000:AF= 1.0000:GF= 1.0000: New in this interaction is that the more diffusible Wnt (= d) has a feedback on the competence f; Higher f, in turn, leads to an increase in the rate of ß-catenin stabilization, causing an advantage of the side that was originally closer to the head and orienting regeneration. Other components as in Fig. 5.

  18. Signaling for head, foot and tentacle formation (Fig. 7): Pattern formation during growth Both the head activator and the foot activator have a long-ranging positive feedback on the competences, i.e., on the ability to perform the pattern-forming reaction. Both competences mutually down-regulate each other. Note that it is highly non-trivial that a polar pattern can be generated at a small size and growth is possible without loosing the polar character. Green: head signal, pink: foot signal; brown: tentacle signal Blue: competence to generate the head signal; gray: competence for the food signal

  19. Signalling for head, foot and tentacle formation(Fig. 7) Regeneration in a near-head fragment In a fragment, all structures reappear at the correct position. In a near-head-fragment, the signal for tentacle formation can regenerate directly since the competence for head signaling (blue) is high enough. The transient tentacle signal at the tip becomes displaced to the final position by the proper head signal Green: head signal, pink: foot signal; brown: tentacle signal Blue: competence to generate the head signal; gray: competence for the food signal

  20. Program-code and parameter for the head-foot tentacle formation (Fig. 7) FOR i = ja TO js: GOSUB olddecay aq = c * s * (A * A + ba) ‘head competence c is a factor in the axt(1, i) = olddecaydiffA + aq / B / (1 + sb * d) ‘ selfenhancement axt(2, i) = olddecaydiffB + aq + bb axt(3, i) = olddecaydiffC + sc * A + bc - cc * c * f ' Head-competence dq = f * rd * arandomxt(i) * (d * d + bd) ' Foot activator axt(4, i) = olddecaydiffD + dq / e /(1 + sf * a) axt(5, i) = olddecaydiffE + dq + be ' foot inhibitor axt(6, i) = olddecaydiffF + rf * d + bf - cf * c * f ' foot-competence gq = c * rg * (g * g + bg) / (1 + sg * g * g) / (1 + ce * A) axt(7, i) = olddecaydiffG + gq / zh ' tentacle activator axt(8, i) = olddecaydiffH + gq + bh ' tentacle inhibitor NEXT I KT= 60:KP=2000:KX= 1:KY= 5:KD= 19:KI= 38:KE= 31:KR= 1:KN= 8:KG= 2: K1= 2:K2= 3:K3= 0:K4= 0: DA= 0.0020:RA= 0.0030:BA= 0.0500:SA= 0.0000:CA= 0.0000:AA= 1.3000:GA= 1.0000: DB= 0.2000:RB= 0.0060:BB= 0.0000:SB= 0.0000:CB= 0.0000:AB= -0.0020:GB= 0.2000: DC= 0.0030:RC= 0.0001:BC= 0.0001:SC= 0.0001:CC= 0.0060:AC= -0.0300:GC= 0.2000: DD= 0.0015:RD= 0.0030:BD= 0.1000:SD= 0.0000:CD= 0.0000:AD= 1.0000:GD= 1.0000: DE= 0.2000:RE= 0.0060:BE= 0.0001:SE= 0.0000:CE= 0.3000:AE= 0.2000:GE= 0.2000: DF= 0.0020:RF= 0.0001:BF= 0.0001:SF= 0.0000:CF= 0.0060:AF= 0.2000:GF= 0.2000: DG= 0.0050:RG= 0.0200:BG= 0.0100:SG= 0.0100:CG= 0.0000:AG= 0.0000:GG= 0.0000: DH= 0.2000:RH= 0.0300:BH= 0.0010:SH= 0.0000:CH= 0.0000:AH= 0.0000:GH= 0.2000:

  21. Signalling for head, foot and tentacle formation (Fig. 7): Regeneration in a near foot fragment In a near-foot fragment (or during budding) the competence to form the head-generating signal is too low to trigger tentacle formation directly. Only after increase o the head competence under the influence of the head signal, tentacle formation is possible. The tentacle signal appears directly at the appropriate position, in agreement with the observation. Green: head signal, pink: foot signal; brown: tentacle signal Blue: competence to generate the head signal; gray: competence for the food signal

  22. Signalling for head, foot and tentacle formation (Fig. 7): Double heads after increase of competence If the head competence is increased everywhere, e.g., by treatment with drugs, ectopic heads and tentacles are formed. After foot removal, a second head my regenerate instead. After washing out of the drug, a new food forms in the centre. Green: head signal, pink: foot signal; brown: tentacle signal Blue: competence to generate the head signal; gray: competence for the food signal

  23. Program-code and parameter for pre-head signal formation (Wnt2) on a cylinder (Fig. 8) CASE 32 'HY- - HYDRA HYPOSTOME, FOOT WITH OWN COMPETENCE, PRE-HEAD SIGNAL FOR iy = kay TO jy: GOSUB leftb FOR ix = kax TO js: GOSUB afcstore aq = c * s * (A * A + ba + ch * g) ‘ Activator, pre-head (g) has activating influence axy(1, ix, iy) = olddecaydiffA + aq / B /(1 + sb * d) ‘ head activator, inhibited by foot (d) axy(2, ix, iy) = olddecaydiffB + aq + bb ‘Long-ranging head inhibition axy(3, ix, iy) = olddecaydiffC + sc * A + bc - cc * c * f ‘Head-competence dq = f * rd * arandom(ix, iy) * (d * d + bd) ‘Foot signal axy(4, ix, iy) = olddecaydiffD + dq / e /(1 + sf * a) ‘Foot signal axy(5, ix, iy) = olddecaydiffE + dq + be ‘Foot inhibitor axy(6, ix, iy) = olddecaydiffF + rf * d + bf - cf * c * f 'Foot-Source gq = rg * arandom(ix, iy) * (g * g + bg) axy(7, ix, iy) = olddecaydiffG + gq / zh / (1 + cg * f + sg * b) 'pre-head-bud axy(8, ix, iy) = olddecaydiffH + gq + bh ‘pre-head inhibitor NEXT: NEXT 2-KT 1000-KP 8-KX 25-KY 7-KD 36-KI 32-KE 1-KR 8-KN 0-KG 2-K1 3-K2 4-K3 0-K4 6.00-DX 0.00-DY 12.00-DZ gadcf-DW 0.0020 0.0030 0.0500 0.0000 0.0000 1.3000 1.0000 0.2000 0.0060 0.0000 0.0000 0.0000 0.0000 0.2000 0.0030 0.0001 0.0001 0.0001 0.0060 -0.0300 0.5000 0.0015 0.0030 0.0500 0.0000 0.0000 1.0000 1.0000 0.2000 0.0060 0.0000 0.0000 0.3000 0.2000 0.2000 0.0020 0.0001 0.0001 0.0000 0.0060 0.2000 0.2000 0.0020 0.0015 0.0080 1.0000 0.3000 0.0000 0.0000 0.2000 0.0020 0.0002 0.0000 0.1500 0.0000 0.2000

  24. Initiation of the pre-foot ring at a distance of the pre-head signal (Fig. 9) Wnt3 At low competence (blue, source density), first the pre-head signal (red, e.g., Wnt2) triggers. In turn, Wnt2 triggers ring-shaped ‘pre-foot’ signal (black). Increasing source density under Wnt2/Wnt3 influence leads to the replacement of Wnt2 by Wnt3. The Pre-foot signal does not depend on Wnt3 and can remain in place pre-head, e.g., Wnt2 pre-foot Source density A non-trivial connection: for maintaining the pre-foot ring without disintegration as occurring in tentacle formation ring formation self-enhancement has to saturate; saturation in turn, allows shift of activation

  25. Program-code and parameter for pre-food signal formation during budding (Fig. 9) CASE 201 'HY- - OLDER HYDRA WNT" AND WNT 3 ATTEMPT TO SWITSCH ' e => diffusible, made by W2 to trigger foot at a distance; FOR iy = kay TO jy: GOSUB leftb FOR ix = kax TO js: GOSUB afcstore aq = c * s * (A * A + sb * d + ba) 'for Head-activator Wnt3a (green) dq = cd * arandom(ix, iy)* (d * d + bd) 'for pre head activator Wnt2 (red) axy(1, ix, iy) = olddecaydiffA + aq / b 'Head inhibitor axy(2, ix, iy) = olddecaydiffB + aq + dq 'head and pre-head inhibitor axy(3, ix,iy) = olddecaydiffC + cc * (a + d) + bc - cf * c * f 'competence (blue) axy(4, ix,iy) = olddecaydiffD + dq / b /(1 + ce * a * a) 'pre-head signal axy(5, ix, iy) = olddecaydiffe + be * d 'for trigger of Pre-Foot fq = rf * (f * f + bf * e) 'f = axy(6...) = pre-foot axy(6, ix, iy) = olddecaydiffF + fq / (1 + sg * (d+a) + sf * f * f) / g axy(7, ix, iy) = olddecaydiffG + fq + bg 'pre-foot inhibitor NEXT: NEXT KT= 60:KP= 160:KX= 35:KY= 35:KD= 3:KI= 12:KE=201:KR= 1:KN= 7:KG= 0: K1= 0:K2= 0:K3= 0:K4= 0: DA= 0.0020:RA= 0.0030:BA= 0.0000:SA= 0.0000:CA= 0.0000:AA= 0.0000:GA= 0.0000: DB= 0.2000:RB= 0.0040:BB= 0.0000:SB= 2.0000:CB= 0.0000:AB= 1.4000:GB= 0.4000: DC= 0.0070:RC= 0.0001:BC= 0.0000:SC= 0.3000:CC= 0.0001:AC= 0.0150:GC= 0.4000: DD= 0.0020:RD= 0.0030:BD= 0.0000:SD= 0.0100:CD= 0.0020:AD= 2.0000:GD= 0.0000: DE= 0.0500:RE= 0.0030:BE= 0.0200:SE= 0.0000:CE= 0.0000:AE= 0.0000:GE= 0.0000: DF= 0.0010:RF= 0.0020:BF= 0.0100:SF= 0.2000:CF= 0.0000:AF= 0.0000:GF= 0.0000: DG= 0.2000:RG= 0.0030:BG= 0.0001:SG= 0.2000:CG= 0.0000:AG= 0.0000:GG= 0.1000:

  26. Program-code and parameters for supernumerary tentacle formation after Alsterpaullone treatment (Fig. 11) Alsterpaullone stabilizes ß-catenin. Assumed in the simulation an increase in the competence. Tentacles form first at some distance from the original tentacles Green: head signal, pink: foot signal; brown: tentacle signal Blue: competence to generate the head signal

  27. Program-code and parameter: supernumerary tentacle formation after Alsterpaullone (Fig. 11) FOR iy = kay TO jy: GOSUB leftb FOR ix = kax TO js: GOSUB afcstore aq = c * s * (aqf + ba) axy(1, ix, iy) = olddecaydiffA + aq / b ' head activator axy(2, ix, iy) = olddecaydiffB + aq + bb ' head inhibitor 'Source density or competence: axy(3, ix, iy) = olddecaydiffC + rc * a + bc - sc * c * f dfq = d * d + bd ‘ for tentacle activator dq = rd * c * arandom(ix, iy) * dfq / (1 + sd * dfq) / (1 + ce * a) axy(4, ix, iy) = olddecaydiffD + dq / e ‘arandom(ix, iy) around 1 with KR % fluctuation axy(5, ix, iy) = olddecaydiffE + dq + be 'tentacle inhibitor ' for foot-system: high head competence (c) is bad for foot activation fq = rf * (f * f + bf) / c axy(6, ix, iy) = olddecaydiffF + fq / g 'foot activator axy(7, ix, iy) = olddecaydiffG + fq + bg 'foot inhibitor NEXT: NEXT During simulation bc is changed from .0001 to .0005 (increasing competence) KT= 12:KP=1000:KX= 33:KY= 28:KD= 7:KI= 36:KE= 36:KR= 1:KN= 7:KG= 0: K1= 2:K2= 3:K3= 4:K4= 0: DA= 0.0015:RA= 0.0030:BA= 0.0500:SA= 0.0000:CA= 0.0000:AA= 1.0000:GA= 1.0000: DB= 0.2000:RB= 0.0040:BB= 0.0000:SB= 0.0000:CB= 0.0000:AB= 1.4000:GB= 0.2000: DC= 0.0020:RC= 0.0001:BC= 0.0001:SC= 0.0001:CC= 0.0000:AC= -0.0500:GC= 0.2000: DD= 0.0050:RD= 0.0200:BD= 0.0050:SD= 0.0100:CD= 0.0200:AD= 0.0000:GD= 0.0000: DE= 0.2000:RE= 0.0300:BE= 0.0040:SE= 0.0000:CE= 0.3000:AE= 0.0000:GE= 0.1000: DF= 0.0030:RF= 0.0020:BF= 0.0100:SF= 0.0100:CF= 0.0000:AF= 0.0000:GF= 1.0000: DG= 0.2000:RG= 0.0030:BG= 0.0010:SG= 0.0000:CG= 0.0000:AG= 0.0000:GG= 1.0000:

  28. Some links The program codes are from programs compiled with the FreeBasic. compiler This open-source compiler can be obtained form: http://www.freebasic.net/get (for Windows and Linux). A simple but complete program with many comment lines can be obtained from our website: http://www.eb.tuebingen.mpg.de/departments/former-departments/h-meinhardt/biuprog.html The program shows the generation of a polar, periodic and oscillating patterns; it can be modified and recompiled with this FreeBasic compiler. There also some instructions how to use the compiler and how to obtain a comfortable working environment. A large program package that allows the change of parameters during a session and manipulations like transplantation and cutting is available on the CD accompanying the book “The Algorithmic Beauty of Sea Shells” (Springer, Heidelberg). It also contains the graphic subroutines that were used to plot the figures of the present paper.

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