A Model That Reproduces an Empirical Rule Distinguishing the Developmental Path of Complete Hox Gene Clusters From Split Clusters

1. Introduction

Hox genes play an important role in the development of most animals and plants. Some Hox genes form clusters which are crucial for the Embryogenesis of Metazoa. The importance of this clustering was first noticed by E.B. Lewis who studied the genetics of Drosophila [1]. In 1978, Lewis observed that some genes of the genome (later coined Hox genes) were located in order along the telomeric to centromeric direction as (Hox1, Hox2, Hox3,…). Lewis noticed that in the Hox gene clusters the genes are sequentially ordered and they are expressed in the same order along the antero-posterior axis of the Drosophila embryo [1]. This is an astonishing event since this correlation occurs between extremely distant locations - the genetic sequence in the cell nucleus in one hand and the Drosophila embryo in the other. These two local domains are about 4 orders of magnitude apart from each other. Biomolecular interaction alone  cannot create such correlations [2]. This surprising phenomenon was named Spatial Collinearity (SC). Some years later, another collinearity was observed particularly in the Vertebrates: temporal collinearity (TC). According to TC, the first Hox gene (Hox1) of the Hox cluster started being expressed. Later, Hox2 was expressed and even later Hox3 followed until all Hox genes were expressed following the sequence Hox1, Hox2, Hox3,…[3].

In order to explain these phenomena, a biophysical model (BM) was proposed in 2001 according to which, pulling physical forces can justify the data [2,4]. Several experimental findings were successfully compared to the BM predictions [5,6].

A simple heuristic expression for these pulling forces F was proposed [7].

F = N * P

(1)

In the above equation, the pulling physical forces F are located opposite the telomeric end of the Hox cluster (Figure 1). In Eq. (1), F stands for a quasi - Coulomb force where, for simplicity, the relative distance between the electric charges is omitted [8]. It turns out that this omission reflects a deep connection to the fundamental phenomenon of Symmetry (see the Appendices). Ν and P are standing for the negative and positive charges acting on a Hox cluster. In the above heuristic formulation, the Hox cluster consists of a deployed finite sequence of Hox genes along the telomeric to centromeric ends of the cluster (Hox1, Hox2, Hox3,…). The numbers assign the gene order in the cluster. These numbers determine the order membership to the evolutionary Paralogy Group (PG). (Here is followed Duboule’s definitions of PG [9]).

The contemporary cephalochordate Amphioxus is a descendant of the Amphioxus considered the ancestor of both Drosophila and vertebrates [9]. Amphioxus lived even before the Cambrian period of evolutionary explosion 500 million years ago (Mya). Vertebrates and Drosophila appeared a few Mya later. Amphioxus has 14 Hox genes whereas vertebrates and Drosophila have 13 (Hox14 is missing).

As mentioned above, N represents the microscopic contribution to F and it is a real entity – the negative electric charge of the DNA sequence. P represents a positively charged molecular structure located opposite the telomeric end of the Hox cluster (Figure 1). Contrary to N, P is a fictitious entity as yet, standing for the embryonic-macroscopic contribution to F. The known morphogens of the present time like Sonic Hedgehog, Fiber growth factors, Retinoic acid and the plethora of other morphogenetic factors were fictitious fifty years ago. The existence of P does not contradict any first principle so it is legitimate to anticipate its existence as advocated in [4]. F pulls the Hox genes sequentially out of the cluster (Figure 1). Eq.(1) is a heuristic expression that was successfully tested in several experiments [5 - 8].

Hox genes control the normal development of animals (wild type). Spontaneous mutation of these genes cause severe malformations (Homeosis), consisting of parts of the animal growing in the wrong location of the body. In Homeosis PG ordering is violated.

About twenty years ago, an important advancement was achieved concerning the transfer of specific molecules from outside the cell into the inner domain of the cell nucleus [10 -13]. For example, it was noticed that significant amounts of Activin are gathered outside the cell nucleus. Controlled amounts of this activin were transduced inside the nucleus causing specific modifications on the genome). It is assumed that BM combined with the action of transduction technology can affect Hox gene expression. This possibility is incorporated in the present hypothesis.

BM is based on the hypothesis that pulling forces are applied at the telomeric end of the Hox cluster. This hypothesis was elaborated in detail and it was concluded that the cluster is elongated along the direction of the force [14]. This BM prediction was later confirmed [15-17]. In some cases the measured elongation of the activated Hox cluster was five times longer than the length of the inactive Hox cluster [16]. When Hox cluster activation is initiated, a weak force (F1) pulls the first gene of the cluster (Hox1) out of its niche toward the interchromosome domain (ICD) (Figure 2).

Particularly Hox1 is directed towards the transcription factory domain (TFD) where Hox gene activation (expression) is possible [18], [19]. When the pulling force increases to F2, Hox2 is extruded from its niche. This process continues until all Hox genes are transferred in the TF. For the efficient function of an elongated elastic spring, besides the pulling force at one of its ends, a proper fastening should be applied at its other end. Accordingly, the Hox cluster should be fastened at the centromeric end (Figure 2).

The Vertebrate Hox clusters comprise four homologue clusters (HoxA, HoxB, HoxC , HoxD) as shown in Figure 3 [11]. Each homologue cluster is included in a separate chromosome. In these homologue clusters the PG identity is conserved. However, in the course of Evolution, modifications of the mouse genes are possible up to the point of gene deletion.This ordered mouse Hox clusters remind of a ratchet allowing motion in one direction only [8]. Note that some ‘teeths’ of the ratchet may be missing.

2. Symmetries entangled with gene ordering in Hox gene clusters

Symmetry is an important concept in Science as stressed in [20 - 21] (see also the Appentices below). Self-similarity is the particular symmetry of objects which, although different, they look the same if depicted under a suitable scale unit. Such objects are the fractals where the part looks like the whole with a typical example being the Barnsley fern shown in Figure 4 [22]. Self-similarity is a continuous symmetry extending to all geometric scales. In contrast, the linear ordering (Hox1, Hox2,…Hox13) of the Hox genes of a Hox cluster and the ontogenetic units of the embryo along the AP-axis refer to only two geometric scales. In this spirit, the two entities can be considered as defective self similar [8, 21]. In this defective self-similarity, the PG ordering is conserved where some Hox genes of the cluster may fade out up to extinction during the Whole Genome Duplication D of the evolutionary process [20-21].

Besides this ordering of Hox genes on a finite straight line, in many early larva embryos (e.g. the echinoderms) a circular ordering is superimposed as shown in Figure 5 [23].

Surprisingly, a self-similarity rule applies so that a circular ordering of the Hox cluster corresponds to the embryonic circular ordering as shown in Figure 6. (See details in [8] and [23]).

In Figure 6B the two ends of the cluster are attached to the 3’ and 5’ ends of the flanking chromosome. If the 3’ end of the flanking chromosome is attached to Hox1(and 5’ to Hox13) no novelty is created and A.planci Hox gene order gene is reproduced. In contrast, if the 5’ end of the flanking chromosome is attached to Hox1, Hox2. Hox3 (shown in Figure 6C) a novelty is created. A second breaking follows leading to a new Hox gene order which corresponds to the Hox gene order of the sea urchin [23].

The circular Hox gene clusters can be incorporated in the flanking DNA sequence of the genome. A recent review by T. Hanscom refers to the well known technique of double strand break (DSB) [24]. In the above review, besides the usual medical applications of the DSB methodology, it is extensively emphasized the novel trends of research to explore how DSB can leverage genome evolution. It is here schematically depicted the incorporation of a Hox cluster in the flanking genome (Figure 7).

In (A) the inserted graft in the middle follows the orientation T→ C In (B) the inserted graft in the middle follows the reverse orientation (T ← C).

3. Complete vs split Hox clusters

Hox Gene Collinearity (both Spatial and Temporal) has been unequivocally confirmed in the Vertebrates. However, in recent years it was found that in many other animal species this is not true, particularly in invertebrates. For instance, it was observed that Hox Colinearity is violated in the lophotrochozoa and this violation was associated with the brachiopods whose Hox cluster is broken [25]. In brachiopods both spatial and temporal collinearities are violated,while lophotrochozoan morphological novelties result from Hox Collinearity violation [25]. It was argued above that for the insertion of a circularly organized Hox cluster in the flanking genome, a break (split) of the cluster is necessary as shown in Figure 6C. It is clear that Hox cluster splitting is a necessary step for evolutionary novelties.

It has been emphasized that tight Hox clustering is lost during Evolution [26-29]. More specifically D. Ferrier and P. Holland assumed that Hox clusters are necessarily constrained in their gene order by TC. Moreover, when TC is no more needed, ‘Hox gene clusters may fall apart’ [26, 27]. Similar arguments were put forward before and after the above assumption [28,29].

The Drosophila Hox cluster has 13 genes contained in the two subclusters ANT-C and BX-C depicted in Figure 8 [27]. Drosophila has a typically split Hox cluster. It was shown that this cluster consists of two subclusters Ant-C and BX-C. Drosophila together with the vertebrate Hox cluster originate from a large ancestral Hox cluster [27]. Cloning had later identified Amphioxus as the common ancestor of insects and vertebrates [28], and a one-to-one correspondence between the Amphioxus Hox genes and the Drosophila Hox genes was confirmed [27]. However in this correspondence some Drosophila Hox genes of the ANT-C subcluster developed novel evolutionary non-Hox functions. For instance, the Drosophila complex of Hox genes (zen1, zen2, bcd) corresponds to the ancestral Amphioxus Hox3 gene. Some Central genes have evolved from tandem evolutionary duplications [27]. (see in [27] Figure 1). The BX-C subcluster consists of the 3 last genes (Ubx, Abd-A, Abd-B) of the Drosophila Hox cluster. The summarizing conclusion from the above analysis is that TC is responsible for a complete Hox cluster. If this is not possible (or not needed) the Hox cluster is split. In any case SC is a necessity for a Hox cluster [26].

An Empirical Rule (ER) formulated by D. Duboule is a useful guide [28]: ‘A complete cluster controlling development in time along the Anterior-Posterior axis is non-split, whereas animals developing according to a time-independent mechanism to produce their main body axis are licensed to split their clusters…’ In the framework of BM, a ‘prediction in retrospect’ was proposed according to which TC really disappears and not as if it were real [30]. This disappearance occurs if an extended upstream DNA domain of the Hox cluster is cut-off [31]. In this spirit it is shown in Figure 9 how a split Hox cluster approaches sequentially in time (t1, t2, t3) the state of completeness where TC is no more needed. In this interpretation, the pulling forces of BM provide a strong theoretical support of the Empirical Rule.

In Figure 2D, an alternative situation is shown: if the Hox cluster is not fastened at all, a minimal pulling force can slide the cluster in the transcription factory domain. In the evolutionary course, a spontaneous mutation may occur which dismantles the fastening of the Hox cluster with dramatic repercussions on both evolution and genetic structure and function of the cluster: the disappearance of TC. Therefore, the loss of TC in Drosophila could serve as a further confirmation of the ER.

4. Discussion and Predictions

4.1. Development in the secondary developmental axis.

In chick limbs, the apical ectodermal ridge (AER) controls development responding to morphogen Fibroblast Growth Factor (FGF). If the ectodermal ridge is excised, Hox13 (the last gene of the cluster) switches off. The results from this experiment are illuminating [32]: Hox13 expression can be initiated again (in the absence of the ridge) if beads soaked in FGF are implanted distally. This occurs after a fixed time interval. If the FGF dose is increased the Hox13 rescue occurs earlier. Furthermore, the rate of Hox13 spreading is faster initially and slower at later stages - a sign that passive diffusion is the main mechanism of signal propagation [32]. In the above chick limb bud experiment in the secondary developmental axis, Hox13 expression is most sensitive to AER excision [32]. However, Hox10 and Hox11 are less sensitive to this excision indicating that TC is not uniform along the developmental axis.

4.2. Development in the mouse primary A-P axis

It is interesting to compare the above limb experimental findings [32] to a similar experiment of upstream DNA excision in mouse embryos as described in [30]. In this excision experiment, TC disappearance was in agreement with the BM pulling forces model. This disappearance was based on a different interpretation of the crucial Kondo and Duboule excision experiment [31]. According to this interpretation it is eventually expected TC to reappear. This expectation remains to be tested [30]. To this end it was proposed the reverse path- the insertion of TGF-beta signals. (A detailed description is included in [30]). The proposed disappearance experiment and the eventual reappearance of TC remains unfinished [30]. The direct course of disappearance is confirmed but the palindromic cource of reappearance remains to be tested and the experimental evidence will be decisive [30].

4.3. Quantitative Collinearity

Besides Spatial and Temporal collinearities, a third collinearity – the Quantitative Collinearity (QC) has been also traced. This was a puzzling issue for a long time. A solution was proposed in the framework of BM [4]. QC is determined following two directions. First is the direction along the Time-arrow (Figure 9) and second is the expression intensity along the anterior- posterior axis. For the HoxD expression of a sequence of cells along the horizontal dashed line, the intensity is stronger at the posterior side [4]. The intensity at any point depends on its distance from the morphogen source. It turns out that Diffusion is the main signal propagating mechanism whose spreading velocity depends on the vicinity to the morphogen source [4]]. It was estimated that this velocity is higher near the morphogen source compared to the velocity at a distant location [32]. Consequently, for the HoxD cluster the expression intensity increases following the order HoxD10….HoxD13 (Figure 9) [4]. Note that this is the same mechanism applied to the Morphogen Fibroblast Growth Factor in the chicken limb bud [32], but the morphogen source is at the tip of the bud in the AER. A similar mechanism applies for the expression of split clusters as mentioned in Section 3.

4.4. Scarcity of developmental mechanisms

In Figure 9 the split Hox cluster activation is depicted together with the quantitative collinearity. It is strange to relate so divergent phenomena. This hints at a scarcity of the developmental mechanisms. Is this an evolutionary advantage or disadvantage? It is surely an evolutionary advantage since it is applied to several other instances facilitating Hox activation. For instance, in the case of limb bud development, Hox clusters are activated via an analogue mechanism. Note however that the morphogen source is located at a different area- the AER in the distal tip of the bud [32].

5. Conclusion

New technological advances (e.g. STORM- the stochastic optical reconstruction microscopy) made possible the measurement of quantities and properties that were inaccessible before. Physical tension in Hox clusters is such a case and more specifically the tension of DNA topological domains which are important for Hox gene activation. A.R. Amàndio et al. have recently measured mouse HoxD clusters under physical tension [33]. The origin of this tension is elusive. This team has even considered the possibility of physical forces of BM to be responsible for this phenomenon. In this case they argue that ‘the forces would be generated by the local chromatin interactions themselves, rather than through an asymmetrically localized point of attachment to the nuclear environment’[33]. Indeed this is most probable. Moreover, in the framework of BM, it is expected that in the global Hox cluster activation, there appear specific genes of the cluster (e.g. a small DNA strip containing Hox10 and Hox11) which remain inactive for a variable time interval. The Hox cluster is activated in the shadowed circle. The strip is pushed out of this Hox activation domain shown in (b) of Figure 10 [34].

In conclusion in Figure 10 are shown: (a) Hox10,…Hox12 are activated. (b) (Hox10, Hox11) are pushed out of the activation domain. (c) At a later time, (Hox10, Hox11) reenter in the activation domain, according to the BM increased pulling force.

It is interesting that the theoretical prediction of ‘Biophysics preceds Biochemistry’[34] was experimentally confirmed soon after ‘…structural organization of HoxD cluster may predate transcriptional activation…’ [16].

Natural Sciences are connected to the other branches of intellectual activity. A short epistemological overview is included in [40].

Appendix A.3. The quasi-Coulomb force

As mentioned in the Introduction, a heuristic pulling force F of the BM is represented by Eq.(1):

F = N * P

(1)

Eq.(1) has the form of a quasi-Coulomb force. The proper Coulomb force (CF) is characterized by the following equation

CF = (Q1 * Q2)/ R²

(2)

where Q1 and Q2 are the electric charges (positive or negative) and R the distance between Q1 and Q2. CF may be attractive (if both charges are either positive or negative) or repulsive (if one charge is positive and the other negative).

The quasi-Coulomb force F has the form

F = Q1 * Q2

where the dependence on R is missing. The arbitrary absence of Geometry is motivated by sheer simplicity as mentioned in the Introduction. However, it turns out that this simplicity is crucial because it is related to the internal Symmetry. For example, the equations of a dynamic system are invariant under space translations. Noether proved that such a symmetric system is necessarily followed by a conserved quantity - in this case the momentum (see Appendix A.2).

In any measurement, Symmetry in a variable appears when this variable is absent in its constituent equations. In the example below, is followed the reasoning of Iliopoulos [37].

Consider a completely symmetric body (the sphere) in 3D space as described in a Cartesian system of axes (x, y, z) or a Polar coordinates system ( z, φ, θ). Any measurement on the sphere contains the angles θ and φ . It turns out that the equation of the sphere is:

x² + y² + z² = R²

(4)

where R is the radius of the sphere. In this equation, the variables (θ, φ) are indeed missing in agreement with the above symmetry requirement for Eq.(4): no angular dependence is observable [37].

In Eq.(1’) for the quasi-Coulomb force F, the term R^-2 is missing, so F is an even simpler equation than the proper Coulomb force. The meaning of this omission is that the heuristic pulling force of the BM is a quantity independent of the 3D geometric space:

F = Q1 * Q2.

(5)

Note that in this space, the symmetry of F becomes internal, reminiscent of Heisenberg’s internal variable- the isotopic spin (see Appendix A.1)].