Lewis Wolpert takes up Turing‘s ideas experimentally and produces his own mathematical treatment.
Morphogen diffusion with breakdown stationary state with the solution Wolpert, Journal of theoretical biology 25, 1 (1969) Take a source at one end of the embryo and let the morphogen diffuse through it.
Once such a gradient exists, it can be used to encode positional information by increasing the expression of certain proteins.
How morphogens actually work we‘ll see in example 2... But there‘s more: positional information is kept when different genes are expressed – and development is robust (sea urchins always look the same no matter what you take away from them... So there‘s scaling.
Chick limb development: the morphogen sonic hedgehog in the early limb determines the later fate.
A change in morphogens can also change the orientation of a limb
Extremity development is crucially dependent on the right positional information at a very early stage.
More reaction-diffusion systems and more physics: Hans Meinhardt Gierer & Meinhardt, Kybernetik 12, 30 (1972).
Hörstadius & Wolsky, Roux‘ Archives 135 69 (1936). Such activator-inhibitor systems can explain classical polarity experiments. In sea urchins
Müller, Differentiation 42 131 (1990). In Hydra
Such reaction diffusion systems of three different morphogens can also lead to spatial stabilization.
This isn‘t just an academic plaything – the proteins MinC, MinD and MinE, which are important in the division of E. coli show exactly these oscillations. Thus leading to an accurate splitting. Raskin & de Boer, PNAS 96 4971 (1999).
3.6.2 A primer in pattern formation Start with the Gierer-Meinhardt equations as an example: For simplicity, we set a = h = 0
Solve them for the homogeneous steady state (i.e. = 0 and t = 0): Then perturb this state with a harmonic function and only keep terms linear in a 0 and h 0 :
This gives the linear system of equations: with
There is only a solution with non-zero a and h if the discriminant of the Matrix is zero: with
The fluctuations only grow if the real part of > 0. The critical value is thus given by Re( ) = 0. If has complex values (i.e. > ( /2) 2 ), the real part is given by /2 and hence the condition is = 0. Thus
On the other hand, if is real valued, then it is only zero if = 0. This yields: A spatial pattern can therefore only develop in an embryo, if ist size exceeds L c. As long as the length is close to L c, this also implies a polarity, since the cosine does not recover on this length scale.
We can do this more generally by assuming that k is continuous. Then we look at which wave number disturbance grows fastest: while Re( ) > 0
Again we start with the case that w is complex: then Re( ) = - /2 and the fastest growing wavenumber is k = 0. The fact that is complex and that Re( ) > 0 lead to conditions for m where we are in this case of a growing homogeneous state that oscillates.
If is real, we obtain: and is is positive if:
All of this is summarized in the Stability diagram: Homogeneous, static pattern growing, inhomogeneous pattern Oscillating, homogeneous pattern
Another set of differential equations describes a threshold switch
Simulation of animal coatings using reaction diffusion and a switch
lepard cheetagiraffe Simulation results for pigmentation
3.6.3 An example: The anterio- posterior axis in Drosophila. Nüsslein-Vollhard & Wieschaus, Nature 287 795 (1980).
Three different sets of genes Nüsslein-Vollhard & Wieschaus, Nature 287 795 (1980).
So there‘s a hierarchy of genes and proteins in the early development
Or specifically Bicoid binding sites Map of the hb gene indicating the locations of bcd-binding sites. The 2.9 kb hb transcript is expressed in an anterior domain which extends from 55-100% egg length, whereas the 3.2 kb transcript (which is expressed maternally and zygotically) is localized to 0- 25% egg length. A, B and C are the fragments identified in the experiment shown in Fig. 2. b, Nucleotide sequences of the regions where bcd protein binds to hb regulatory regions. Base pairs protected against DNaseI digestion as referred from Fig. 4 are indicated by a bar below the sequence. Driever & Nüsslein-Vollhard, Nature 337 138 (1989).
Lets get back to the development of the anerioposterior axis
The first step is the most important – get a gradient going à la Wolpert! Driever & Nüsslein-Vollhard, Cell 54 95 (1988).
Bicoid is known to act as an activator for hunchback expression. Driever & Nüsslein-Vollhard, Nature 337 138 (1989).
Bicoid protein also has the exponential gradient one expects from a maternally deposited morphogen!
However things are a little more complicated than that – Hunckback is expressed too precise in order to just be determined by a Wolperian Bicoid... Houchmandzadeh, Wieschaus and Leibler, Nature 415, 798 (2002)
Furthermore, the boundary is always in the middle of the embryo irrespective of ist size – this would not be expected from an exponential gradient, which sets a length scale.