We can establish the following correlations between cell locations, Coulomb forces and subsequent Hoxd gene extrusions and subsequent activations.
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1)
For cells expressing only one gene (Hoxd 9):
P = 2, N = 10, F =2 × 10 = 20, L = 1 (Figure 1a), Domain D9
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2)
For cells expressing two genes (Hoxd9 and Hoxd10):
P = 4, N = 10, F =4 × 10 = 40, L = 2 (Figure 1b), Domain D10 < Domain D9
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3)
For cells expressing three genes (Hoxd9, Hoxd10 and HoxdH):
P = 6, N = 10, F = 6 × 10 = 60, L = 3, Domain D11 < Domain D10 < Domain D9...etc.
In experiments where one gene of the subcluster is deleted, the negative charge of the cluster will decrease to N = 8 and the length of the extruded fibre will be shorter. Two gene deletions will produce an even bigger reduction in the length of the extruded fibre. By contrast, when one gene is duplicated, N will increase to 12 and the extruded fibre will be longer etc.
In the following, we estimate how deletions and duplications modify gene expressions, and particularly in domains D9, D10 and D11.
(a) Posterior gene deletions
Consider cells in which Hoxd 9 is normally activated (domain D9). The normal force acting on the fibre is (at least): F1 = 2 × 10 = 20 (Figure 1a). If posterior gene Hoxd10 is deleted, the negative charge N of the cluster decreases to 8 and the pulling force on the fibre becomes F = 2 × 8 = 16, which is below the normal value of 20. Consequently, Hoxd9 will retreat towards the interior of the CT (Figure 1Sa (Figure 2); left).
For the complete extrusion of Hoxd9, a stronger force is needed and this is achieved by a posterior shift to cells where the morphogen value is higher, as is the corresponding P value: P (++ → +++).
If more than one posterior gene is deleted, the posterior shift of Hoxd9 activation will increase. Tarchini and Duboule [4] designated this as an unexpected observation. For the biophysical model, this posteriorisation is an obvious outcome.
(b) Anterior gene deletions
This case is more involved because, besides the charge modification of the cluster, the intervention also affects the length of the extruded gene fibre (Figure 1Sb (Figure 3). Consider, for example, cells in which Hoxd9 and Hoxd10 are activated, where the normal extrusion force is (at least) F2 = 40. If Hoxd 9 is deleted, the extrusion force on the fibre is reduced to F = 4 × 8 = 32, which is lower than 40. In this situation, Hoxd10 is the first extruded gene. For this first gene, we estimated that the necessary extrusion force is F1 = 20 at least.
For activation of Hoxd10, F can be reduced to F1, and this occurs at more anterior cell positions with alower P value: P (++++ → +++). An ectopic expression of Hoxd10 in the anterior region of the limb bud will be observed, associated with a premature expression of the gene. This was verified by Tarchini and Duboule [4].
(c) Posterior gene duplications
Compared to gene deletions, gene duplications have an opposite effect on Hoxd gene expressions. Consider, again, cells in which Hoxd 9 and Hoxd10 are activated, where the pulling force is (at least) F2 = 40. If posterior gene Hoxd11 is duplicated (Figure 2Sa (Figure 2)), the charge of the cluster will increase to N = 12.
The pulling force on the fibre will become: F = 4 × 12 = 48, which is higher than F2. Activation of Hoxd10, however, can start with F2; therefore, F can be reduced to this value. This is achieved by an anterior shift of D10; in this case, the morphogen decreases, and so does P (++++ → +++) and F → F2. The result is an anteriorisation and premature expression of Hoxd10.
Let us examine Hoxd10 expression at a position and time without Hoxd11 duplication. At this position, after Hoxd11 duplication, Hoxd10 will be abnormally translocated further inside the ICD, away from the CT border. A downregulation of Hoxd10 expression is expected at this position (quantitative collinearity). This was verified by Kmita et al. [6]
(d) Anterior gene duplications
Consider the case of two activated genes (Hoxd9 and Hoxd10), where F2 = 40. If anterior gene Hoxd 9 is duplicated, the pulling force on the fibre increases from F2 to F = 48. For activation of Hoxd10, however, this increase is not sufficient, since now three genes must be extruded. The corresponding pulling force for three genes is F3 = 60 (Figure S2b (Figure 3)). For a stronger force F3, P must take a higher value: P (+ + + → + + + + +). Therefore, after an anterior gene duplication, the expression of Hoxd10 will be shifted posteriorly, in agreement with the findings of Tarchini and Duboule [4].
These calculations can be applied to extended gene deletions and duplications incorporating several genes from the entire Hoxd cluster. Following the above rules, the estimation of anteriorisations or posteriorisations is straightforward, and the results agree with the findings of Duboule and co-workers. The conclusion is that Hoxd gene deletions and duplications strongly support the present electric force mechanism of the biophysical model.
Hoxbl
transposition in
Hoxd
Recently, the Hoxb1 transgene was transposed at the 5' end of the Hoxd cluster [8]. The results from this transposition are in agreement with the biophysical model. For example, in the wild-type mouse embryo at stage E7.5, Hoxd13 is not expressed in the primitive streak tissue. By contrast, Hoxd13 is activated when the Hoxb1-LacZ reporter is inserted at the 5' end of the Hoxd cluster. This finding is explained by the biophysical model, since, in analogy with the Hoxd regions, the Hoxb1 transgene carries a negative charge and the total N increases in the transgene embryo (as analysed in detail above). The pulling force on the gene fibre therefore increases, so that Hoxd13 loops out of the CT and its activation is possible. In the wild-type limb bud at E9.5, the Hoxd region, although decondensed, does not extrude out of the CT [8]. In the Hoxb1-LacZ embryo, however, Hoxd does loop out and this can be attributed, again, to the increased force of attraction that pulls the gene fibre out of the CT.