globals [ half-width ; half-width of rows separation ; separation between rows AG-offset ; offset to row with anterior gradient PG-offset ; offset to row with posterior gradient axis-row ; the row on which the axis is displayed AG-row ; the row on which the anterior gradient is displayed PG-row ; the row on which the posterior gradient is displayed A_const ; A gradient space constant P_const ; P gradient space constant head-col ; column position of head P-max ; maximum P signal (decreases when growth limit reached) ] breed [ tailbuds tailbud ] breed [ heads head ] tailbuds-own [ cycle ; phase in clock cycle cell-cycle ; phase in growth cycle ] patches-own [ A_signal ; anterior signal A_new ; new value of A (temp) P_signal ; posterior signal P_new ; new value of P (temp) activity ; S activity (segmentation signal) of an axis patch phase ; phase of axis patch activity future-somite? ; true if destined to become somite somite? ; true if cell is in a somite ] to setup ; Initialize Simulation ca ; compute coordinates of display regions set half-width 4 set separation 1 set AG-offset 1 + separation + 2 * half-width set PG-offset -1 - separation - 2 * half-width set axis-row 0 set AG-row axis-row + AG-offset set PG-row axis-row + PG-offset set head-col min-pxcor + 1 ask patches [ set A_signal 0 set P_signal 0 set activity 0 set phase 0 set future-somite? false set somite? false set pcolor grey + 3 ; light grey ] create-tailbuds 1 [ init-tailbud ] ; one tailbud at right (posterior) end of embryo create-heads 1 [ init-head ] ; one head cell at left (anterior) end of embryo ask patch head-col axis-row [ set somite? true ] ; head acts like somite set P-max 100 ; tail bud emits this so long as growth continues set A_const sqrt (A_diff / (2 * A_decay)) ; space constant of A gradient set P_const sqrt (P_diff / (2 * P_decay)) ; space constant of P gradient display-axis ; display initial state of embryo display-AG ; display 0 A gradient display-PG ; display 0 P gradient end to reset_defaults ; Reset parameters to default values set growth_rate 2000 set clock_period 500 set A_diff 50 set A_decay 10 set P_diff 50 set P_decay 5 set A_upb 70 set P_upb 10 set S_lwb 95 set initial_length 4 set max_length max-pxcor set refractory_period 100 set threshold 10 end to init-tailbud set initial_length min list (max-pxcor - head-col - 1) initial_length ; ensure initial length in bounds setxy (head-col + initial_length) axis-row set shape "circle" set color blue set size 4 set cycle 0 set cell-cycle 0 set heading 90 end to init-head setxy head-col axis-row set shape "circle" set color green set size 4 end to simulate ; Main simulation cycle diffuse-AG diffuse-PG ask tailbuds [ oscillate grow ] ; there is only one tailbud propagate-wave display-AG display-PG display-axis end to oscillate ; Advance cycle of pacemaker with color changes set cycle cycle + 1 ifelse cycle >= clock_period [ set cycle 0 set color orange ; orange indicates S has been emitted set activity 300 ] ; 300 units of S emitted from tailbud [ if cycle > clock_period / 4 ; for visibility, tailbud stays orange for 1/4 cycle [ set color blue ] ] end to grow ; Tailbud grows (moves right) until limit reached ifelse xcor < (min list max_length max-pxcor) - 1 ; grow so long as less than limit [ fd growth_rate / 100000 ] ; grow at specified rate [ set P-max max list 0 (P-max - 1) ] ; P-max decays when limit reached end to propagate-wave ; Implement propagation of S in excitable medium ; activity (S morphogen) at tailbud decays ask patch ([pxcor] of tailbud 0) axis-row [ set activity activity * 0.95 ] ; following addresses all patches on axis between head and tailbud (exclusive) ask patches with [ pycor = axis-row and pxcor >= head-col and pxcor < [pxcor] of tailbud 0 ] [ ; approximate diffusion and decay of S: set activity activity + (([activity / 2] of patch-at -1 0) + ([activity / 2] of patch-at 1 0) - activity) * 0.75 set activity activity * 0.95 if phase >= 0 [ ; REDUNDANT? (always >= 0) ifelse phase = 0 ; refractory period is past [ if activity > threshold [ ; if threshold exceeded, cell fires set phase refractory_period ; enter new refractory period set activity activity + 100 ] ] ; emit pulse of S [ set phase phase - 1 ; activity continues to decay during refractory period set activity max list 0 (activity - int (100 / refractory_period + 1) ) ] ] if not somite? and activity > S_lwb ; four conditions for triggering somite formation and [A_signal < A_upb] of patch-at 0 AG-row and [P_signal < P_upb] of patch-at 0 PG-row [ set future-somite? true ] ] ; following addresses all patches involved in axis display ask patches with [ abs(pycor - axis-row) <= half-width and pxcor >= head-col and pxcor < [pxcor] of tailbud 0 ] [ if [future-somite?] of patch pxcor axis-row ; if the cell on axis will be a new somite [ ; if cell to left is already a somite, then this is anterior boundary of new somite ifelse [somite?] of patch (pxcor - 1) axis-row [ set pcolor yellow ] ; anterior compartment of somite [ set pcolor brown ] ; posterior compartment of somite ] ] ask patches with [future-somite?] [ set future-somite? false set somite? true ] ; future somites commit end to diffuse-AG ; Diffuse anterior morphogen and set to maximum in somites ask patches with [ pycor = AG-row and pxcor > head-col and pxcor < [xcor] of tailbud 0 ] [ set A_new A_signal + (([A_signal / 2] of patch-at -1 0) + ([A_signal / 2] of patch-at 1 0) - A_signal) * A_diff / 100 set A_new A_new * (1 - A_decay / 100) ] ask patches with [ pycor = AG-row and pxcor > head-col and pxcor < [xcor] of tailbud 0 ] [ set A_signal A_new ] ask patches with [somite?] [ ask patch-at 0 AG-row [set A_signal 100] ] end to diffuse-PG ; Diffuse posterior morphogen and set to P_max at tail bud. ask patch ([pxcor] of tailbud 0) PG-row [ set P_signal P-max ] ask patches with [ pycor = PG-row and pxcor > head-col and pxcor < [xcor] of tailbud 0 ] [ set P_new P_signal + (([P_signal / 2] of patch-at -1 0) + ([P_signal / 2] of patch-at 1 0) - P_signal) * P_diff / 100 set P_new P_new * (1 - P_decay / 100) ] ask patches with [ pycor = PG-row and pxcor > head-col and pxcor < [xcor] of tailbud 0 ] [ set P_signal P_new ] end to display-AG ; Display Anterior Gradient ; The lower bound is 1 because 0 seems to cause incorrect colors (black instead of white) ; for certain values near 0 ask patches with [ abs(pycor - AG-row) <= half-width and pxcor >= head-col and pxcor <= [xcor] of tailbud 0 ] [ set pcolor scale-color green ([A_signal] of patch pxcor AG-row) 150 1 ] end to display-PG ; Display Posterior Gradient ; The lower bound is 1 because 0 seems to cause incorrect colors (black instead of white) ; for certain values near 0 ask patches with [ abs(pycor - PG-row) <= half-width and pxcor >= head-col and pxcor <= [xcor] of tailbud 0 ] [ set pcolor scale-color blue ([P_signal] of patch pxcor PG-row) 150 1 ] end to display-axis ; Display segmentation (S) concentration of undifferentiated cells ask patches with [ abs(pycor - axis-row) <= half-width and pxcor >= head-col and pxcor < [xcor] of tailbud 0 ] [ if not [somite?] of patch pxcor axis-row [ set pcolor scale-color orange ([activity] of patch pxcor axis-row) 0 150 ] ] end @#$#@#$#@ GRAPHICS-WINDOW 15 15 669 194 -1 18 4.0 1 10 1 1 1 0 0 0 1 0 160 -18 18 0 0 1 ticks BUTTON 15 210 89 243 default reset_defaults NIL 1 T OBSERVER NIL NIL NIL NIL BUTTON 120 210 186 243 setup setup NIL 1 T OBSERVER NIL NIL NIL NIL SLIDER 15 250 187 283 growth_rate growth_rate 0 5000 2000 1 1 * 10^-5 HORIZONTAL SLIDER 15 290 187 323 clock_period clock_period 0 2000 500 1 1 NIL HORIZONTAL BUTTON 205 490 378 535 run simulate T 1 T OBSERVER NIL NIL NIL NIL SLIDER 15 330 187 363 A_diff A_diff 0 100 50 1 1 % HORIZONTAL SLIDER 15 370 187 403 A_decay A_decay 0 100 10 1 1 % HORIZONTAL SLIDER 15 410 187 443 P_diff P_diff 0 100 50 1 1 % HORIZONTAL SLIDER 15 450 187 483 P_decay P_decay 0 100 5 1 1 % HORIZONTAL SLIDER 205 210 377 243 initial_length initial_length 0 160 4 1 1 NIL HORIZONTAL SLIDER 205 410 378 443 refractory_period refractory_period 0 1000 100 1 1 NIL HORIZONTAL SLIDER 205 450 377 483 threshold threshold 0 100 10 1 1 NIL HORIZONTAL SLIDER 205 290 377 323 A_upb A_upb 0 100 70 1 1 NIL HORIZONTAL SLIDER 205 330 377 363 P_upb P_upb 0 100 10 1 1 NIL HORIZONTAL SLIDER 205 370 377 403 S_lwb S_lwb 0 100 95 1 1 NIL HORIZONTAL MONITOR 15 490 97 535 A const A_const 2 1 11 MONITOR 105 490 188 535 P const P_const 2 1 11 SLIDER 205 250 377 283 max_length max_length 0 160 160 1 1 NIL HORIZONTAL TEXTBOX 390 210 635 351 COLORS:\n\ngreen - A morphogen\nblue - P morphogen\norange - S morphogen\nyellow - anterior boundary of somite\nbrown - posterior part of somite 13 0.0 1 @#$#@#$#@ WHAT IS IT? ----------- Humans have 33 vertebrae, chickens have 55, mice have 65, and the corn snake has 315. How does a developing embryo ÒcountÓ the number of vertebrae it should produce, which is characteristic of its species? This NetLogo model is based on the best contemporary explanation of this morphogenetic (form-producing) process, called the Òclock and wavefront modelÓ of embryological segmentation (Cooke & Zeeman, 1976; see References for recent work). The vertebrae (and muscles and organs associated with them) develop from a series of somites that develop in order from the front (anterior) of the embryo to its tail (posterior). In this process previously uncommitted cells differentiate into somite cells with definite boundaries between distinct somites. The process is controlled by the concentrations of three different chemicals or Òmorphogens.Ó The Anterior or A morphogen (probably retinoic acid in embryos) is produced by the cells that have already committed to be in somites, and it diffuses toward the tail of the embryo, forming a decreasing gradient from the somites to the tail. The Posterior or P morphogen (e.g., FGF, Wnt) is produced by the embryo's tail bud, and it diffuses toward the head of the embryo, forming a decreasing gradient from the tail toward the head. Where the P concentration falls below a certain threshold (called P_upb in this model) is called the Òdetermination front,Ó since somites form in front of it. Since the determination front is at a fixed distance in front of the tail bud, as the embryo grows the determination front moves rearward. As a consequence there is an increasing gap between the determination front and the region of high A concentration near the most recently formed somites. Since this region of low A and P concentrations is where the new somite will form, its size determines the size of the resulting somite, and therefore the embry's growth rate also has an effect on somite size. (The large number of vertebrae in the corn snake results from a comparatively slow growth rate; see Gomez et al., 2008.) The tail bud contains a biochemical Òsegmentation clockÓ that periodically produces a pulse of chemical that is a segmentation signal (Notch & HES proteins, called the ÒS morphogenÓ here); biologists are still elucidating the exact nature of this clock (DequŽant & PourquiŽ, 2008). The clock period is 1.5 hours in chickens, 1.7 hours in corn snakes, and 2 hours in mice, but 4-5 hours in humans. The uncommitted cells form an Òexcitable medium,Ó which means that if the concentration of S around a cell is sufficiently high, the cell is stimulated to produce its own pulse of S. As a consequence, when the clock cells produces a pulse of S, it causes a cascade of S production, which propagates in a wave from the tailbud toward the front of the embryo. Since a cell enters an insensitive Òrefractory periodÓ after it emits a pulse, the wave cannot go backward, but propagates in a single direction (another characteristic of excitable media). When the S (segmentation) signal reaches cells in the region between the determination front and the previously formed somites (i.e., the region of low A and P concentrations), it triggers the cells in this region to commit to being somite cells. Thus the amount of space that has opened between the A and P gradients (as determined by growth) between clock pulses defines the size of the new somite. As the cells commit to becoming somite cells they use local chemical signals to decide whether they are at the anterior or posterior boundary of the somite (so, for example, they know whether to form the anterior or posterior end of a vertebra, thus establishing their polarity). In real embryos, not all the somites are the same size, because the growth rate varies according to the developmental program of the species. HOW IT WORKS ------------ After setup the model displays the initial state of the embryo in a band in the middle, the A gradient in the upper band, and the P gradient in the lower. Furthermore, the central band has a green circle at the left end, representing the embryo's head, and a blue circle at the right end, representing the embryo's tail bud. When the simulation is started, operation proceeds as follows. On each time step the head circle set its A concentration to the maximum 100 and the tail bud sets its P concentration to the maximum (P-max, initially 100). One-dimensional diffusion and decay of the A and P morphogens is simulated, and the resulting gradients are displayed. The embryo grows to the right at the specified rate, and as it moves it sets the S morphogen of its patch to the maximum value (300) whenever its clock resets, which decays at a fixed rate. The axis cells implement an excitable medium. The S morphogen diffuses and decays at fixed rates, but if it exceeds a specified threshold in a cell that is not in its refractory period, then that cell will ÒfireÓ by adding 100 to its S concentration and enter its refractory period. Due to the refractory property, the wave of S activity tends to propogate in one direction, from the tail bud toward the head. In order for a patch to differentiate into a somite, both the A and P morphogens must be less than specified upper bounds (A_upb, P_upb), and the segmentation morphogen must be above a threshold (S_lwb). Normally this occurs in a band to the right of the last differentiated somite, and so this is where the new somite forms. When a patch differentiates into a somite it looks at the patch to its left (anterior side). If it is already a somite, then the patch differentiates into an anterior boundary cell (yellow) of the new somite, otherwise it differentiates into a posterior compartment cell (brown). Thus the yellow-brown segments are the somites. When the embryo reaches its growth limit, the P-max value begins to decay toward 0, and so the P gradient eventually decays to 0. Although the tail bud pacemaker continues to cycle, the segment structure is stable. HOW TO USE IT ------------- DEFAULT sets the parameters to default values at which segmentation occurs. They are a good place to begin investigation of the effects of the parameters. SETUP initializes the simulation based on the specified parameters. (Most parameters can be changed during a simulation run, however.) RUN starts the simulation running. Growth_Rate sets the number of fractions of a unit of length (one patch) that the embryo grows during each clock tick. Clock_Rate defines the number of ticks between clock pulses from the tail bud. A_diff, A_decay, P_diff, P_decay set the diffusion rate and decay rate for the anterior and posterior morphogens. Each is defined as a percentage of the quantity at a patch that is distributed to adjacent patches (for diffusion) or eliminated (for decay). To help in setting these parameters, monitors A_const and P_const display the length constant determined by the diff/decay ratio. The length constant is the length, in patch units, over which the concentration decreases to 1/e = 37% of its maximum value. (The diffusion rate determines how quickly the gradient is established.) Initial_Length, Max_Length are the initial and maximum length (in patch units) of the embryo. When it reaches the maximum length (or the right-hand boundary of the display), it stops growing. A_upb, P_upb, S_lwb sets the morphogen thresholds at which cells will differentiate into somites. The segmentation signal must be greater than S_lwb and the anterior and posterior gradients must be below A_upb and P_upb, respectively, in order for differentiation to occur. This occurs in the region between the already differentiated somites and the determination boundary when a segmentation peak passes through. Refractory Period is the period during which undifferentiated cells are insensitive to S (the segmentation morphogen) after they have emitted a pulse of S. Thus the Refractory Period determines the maximum wave frequency. Threshold is the minimum S (segmentation morphogen) level in adjacent cells in order for a cell to fire and emit its own pulse of S. THINGS TO NOTICE ---------------- [This section will some ideas of things for the user to notice while running the model.] THINGS TO TRY ------------- [This section will give some ideas of things for the user to try to do (move sliders, switches, etc.) with the model. This is all there is for now.] With the clock_period set to 500, try different growth_rates from 500 to 5000 and oberve how it affects the size and number of segments. Mutations can effect the stability of a morphogen and the concentrations of chemicals that break it down, thus altering the decay rate. Other mutations can alter morphogens so as to affect their diffusion rates in embryonic tissue. Keeping in mind that the space constant of the gradient is determined by the diff/decay ratio, experiment with the effects of different diffusion and decay rates on the pattern of segmentation. To observe the wave propagation of the S (segmentation) signal, do the following. Start with the Default parameters and set A_upb and P_upb to 0; this will prevent somite formation. Set the growth_rate to 0 and the initial_length to something large, such as 100. This will allow you to observe the wave propagating from the tail to the head of a relatively large, fixed-size embryo. When you Run the simulation, watch the tail bud. When it turns orange, it has generated a pulse of S. The leading edge of the wave is orange, showing a high S concentration, while the trailing edge decays back to black. Try a variety of different clock_periods longer than the refractory_period. Then try setting the clock_period to less than the refractory_period. Observe carefully what happens and try to explain it. EXTENDING THE MODEL ------------------- This section will give some ideas of things to add or change in the procedures tab to make the model more complicated, detailed, accurate, etc. NETLOGO FEATURES ---------------- This model implements approximate one-dimensional diffusion directly, rather than using NetLogo's built-in diffusion commands, which are two-dimensional. Agent sets (in particular patch sets) that vary with the growing embryo are used to address the patches in the various display regions and the patches involved in diffusion and wave front computations. The scale-color operators have a lower bound of 1 (rather than 0), because a lower bound of 0 caused artifacts to appear in the gradient display. It appears that certain values very near to 0 (and thus colors that should have been near white) were translated to black. CREDITS AND REFERENCES ---------------------- For recent research on segmentation in embryological morphognesis, on which this model is based, see: DequŽant, M.-L., & PourquiŽ, O. (2008). Segmental patterning of the vertebrate embryonic axis. Nature Reviews Genetics 9: 370Ð82. Gomez, C., …zbudak, E.M., Wunderlich, J., Baumann, D., Lewis, J., & PourquiŽ, O. (2008). Control of segment number in vertebrate embryos. Nature 454: 335Ð9. The original statement of the Òclock and wavefront modelÓ is: Cooke, J., & Zeeman, E.C. (1976). A clock and wavefront model for control of the number of repeated structures during animal morphogenesis. J. Theor. Biol. 58: 455Ð76. To refer to this model in academic publications, please use: MacLennan, B.J. (2008). NetLogo Segmentation model. http://www.cs.utk.edu/~mclennan. Dept. of Electrical Engineering & Computer Science, Univ. of Tennessee, Knoxville. In other publications, please use: Copyright 2008 Bruce MacLennan. All rights reserved. See http://www.cs.utk.edu/~mclennan/420/NetLogo4.0/Segmentation.html for terms of use. @#$#@#$#@ default true 0 Polygon -7500403 true true 150 5 40 250 150 205 260 250 airplane true 0 Polygon -7500403 true true 150 0 135 15 120 60 120 105 15 165 15 195 120 180 135 240 105 270 120 285 150 270 180 285 210 270 165 240 180 180 285 195 285 165 180 105 180 60 165 15 arrow true 0 Polygon -7500403 true true 150 0 0 150 105 150 105 293 195 293 195 150 300 150 box false 0 Polygon -7500403 true true 150 285 285 225 285 75 150 135 Polygon -7500403 true true 150 135 15 75 150 15 285 75 Polygon -7500403 true true 15 75 15 225 150 285 150 135 Line -16777216 false 150 285 150 135 Line -16777216 false 150 135 15 75 Line -16777216 false 150 135 285 75 bug true 0 Circle -7500403 true true 96 182 108 Circle -7500403 true true 110 127 80 Circle -7500403 true true 110 75 80 Line -7500403 true 150 100 80 30 Line -7500403 true 150 100 220 30 butterfly true 0 Polygon -7500403 true true 150 165 209 199 225 225 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