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Humans have 33 vertebrae, chickens have 55, mice have 65, and the corn snake has 315. How does a developing embryo "count" 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 "clock and wavefront model" 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 "morphogens." 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 "determination front," 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 "segmentation clock" that periodically produces a pulse of chemical that is a segmentation signal (Notch & HES proteins, called the "S morphogen" here); biologists are still elucidating the exact nature of this clock (Dequéant & Pourquié, 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 "excitable medium," 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 "refractory period" 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.
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 "fire" 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.
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.
[This section will some ideas of things for the user to notice while running the model.]
[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.
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.
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.
For recent research on segmentation in embryological morphognesis, on which this model is based, see:
Dequéant, M.-L., & Pourquié, O. (2008). Segmental patterning of the vertebrate embryonic axis. Nature Reviews Genetics 9: 370-82.
Gomez, C., Özbudak, E.M., Wunderlich, J., Baumann, D., Lewis, J., & Pourquié, O. (2008). Control of segment number in vertebrate embryos. Nature 454: 335-9.
The original statement of the "clock and wavefront model" 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: 455-76.
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.
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
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