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Models of Development
How does structure emerge from a structureless state without an externalorganising force?We all start out as an undifferentiated clump of cells, all withthe same genetic code ?
Dr.S.S.D.PandeyGlobal Synergetic [email protected] 15th June 2008
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Dr.S.S.D.PandeyGlobal Synergetic Foundation
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Self Organisation
Complex patterns can be produced by application of simple rules to
simple initial states.
In biological systems chemical signals combined with nonlinear localinteraction are able to produce very complex patterns.
Symmetry breaking: the initial undifferentiated state isunstable. Feedback amplifies deviations, turning noise intopattern.
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How to Make a Brain
Grow a large number of similar cellsAllow them to differentiate into different types depending on
the neighborhood.Diffuse long range chemical signalsLet neurons send out connections towards their favoritechemical(s).Connect, and possibly remove failed cells and connections.Repeat, add other kinds of plasticity.
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Reaction-Diffusion Systems
An important class of pattern forming systems consists ofchemicals diffusing and reacting with each other.
If the interactions are nonlinear and the diffusion coefficientsdifferent, interesting instabilities result.
Can be modelled with partial differential equations or cellularautomata.
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Turing Patterns
Alan Turing 1952: Spontaneous symmetry breaking inreaction-diffusion systems.
Diffusion of morphogens among identical cells ordered in aRing
Turing showed that when individually stable cells areconnected, diffusion can cause instability and patternformation
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Analysis of linear system
A basic example with two morphogens. Let x and y be theirdeviations from equilibrium in each cell:
In order to get stability of isolated cells (no diffusion terms
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The signs of the constant matrix are forced into one of these forms
Activator and depleted substrate
Activator and inhibitor
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The normal mode trick
The behavior of x and y can be analysed by expressing themas normal modes
ξ and η can now be used instead; this is usually simplerbecause different Fourier modes are decoupled
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Doing the stability analysis for ξ and η gives:
As long as μ and ν are small the steady state x=y=0 is stable.Beyond a certain point, one of the modes will become unstable, andsmall disturbances will create an exponentially growing periodicpattern.Diffusion produces patterns instead of dissolving them!This requires that μ is not equal to ν; lateral inhibition
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Meinhardt: Hydra Development
Model head development as activator-inhibitor pair with characteristiclength of the same order as body length.
Regeneration after division.
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Symmetry breaking during growth
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Effect of cutting
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Diffusion works only over short scales.
Embryonic fields in which developmental decisions take place are smaller 1mm and less than 100 cells across
The early local state is translated into a permanent cell state by changes ingene expression
A hierarchy of genes that activate each other in turn
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Meinhardt: Segmentation
Several morphogens dependent on main gradient.Location information important
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Segmentation as gene activation
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Reparation of a gap
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Reversal
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Hamahashi and Kitano: DrosophilaEmbryogenesis
Reaction diffusion of gene products regulating gene expression
Hierarchical arrangement of maternal, early and late genes
Based on known genes
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Meinhardt: Snail Shells
1D reaction diffusion system.Complex shell patterning involves several
morphogens.
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2D Reaction-Diffusion Systems
Wide variety of patternsLeaf primordia, fur patterns, vasculature, retinotopic maps
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Barrio et. al.: Fish Patterns
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Spots and leaf primordia
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Vascular networks
Cells differentiate when a signal reaches a thresholdSignal production is dependent on a substrate that isremoved by differentiated cells
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Hentschel & Fine:
Activity Dependent Dendritic Morphogenesis
Calcium as a morphogen.Sub membrane calcium concentration controls the local growthvelocity.Excitable membrane. Positive feedback for calcium influx.Calcium catalytic growth produces branching patterns
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Hely, Graham and van Ooyen:
Dendrite branching
Elongation and branching due to MAP2 bound to microtubuli
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AxogenesisAxons seek out their target using growth cones sensing
chemical gradients.
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How far can a gradient guide?
Three constraints:Too low concentration: almost no receptors will be boundToo high concentration: all receptors will be boundConcentration difference across cone must be larger than noise
Relative difference:
Absolute difference:
Results in a maximum distance of 1mm for a diffusing signal,1cm for an optimally distributed substrate-bound signal.
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Fleischer: Axons following chemical gradients
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Bundling Model of Hentschel and van Ooyen(Model I)
Chemoattractant towards target cells
Chemoattractant from growth cones causes bundling
Chemorepellant from growth cones, dependent on target cell
chemoattractant, causes dispersion near target
Cone growth towards chemical gradient.
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Model II
Instead of chemical signals from growth cones, physicalattachment when they get close enough.
Attachment strength depends on chemo attractant fromtarget.
Random axon movement.
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Results
Model I produces nice fasciculation and unbundling near target.Topological Organisation.
Model II does not bundle globally, does not unbundle. Pathfinder cells.
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Phenomenological Development Modeling
Cellular AutomataDiffusion Limited AggregationLindenmeyer-systems
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Conclusion
Complex patterns and Organisation can beachieved by simple rules.
Diffusion of chemo attractants or morphogens caninducelong-range order.
More and more morphogens and genetic
networks have been identified.
A lot of data is waiting for modeling.
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Thank You very much
Dr.S.S.D.Pandey [email protected]
Centre for Studies in Complexity
Global Synergetic Foundation
www.globalsynergetic.org