On a model of pattern regeneration based on cell memory
Some animals, like planarians or hydra, can regenerate their head or tail (feet) or both. These two organs can produce various signals which propagate in other tissues. If these signals interact with each other (activation-inhibition) and if this interaction results in their nonuniform distribution (pattern formation), then it is possible to imagine that they also participate in the regeneration of the lost organ.
The method suggested in this work does not imply the interaction of two (or more) signals. In the minimal configuration it is sufficient to have only one signal. The main assumption of the model is that cells can keep information about the previous value of this signal when it is changed (< 64 bits of information). One testable prediction of such models is that mechanisms that underlie memory in the nervous system may be likewise implicated in regenerative pattern control.
Let us note that regeneration with memory cells acts locally. It does not require interaction of signals from distant organs. Wound healing is an example of regeneration which occurs locally and where only one tissue can be involved. The model based on cell memory suggests the same approach to wound healing and to regeneration of a single organ or several organs simultaneously.
Interestingly, some species such as planaria possess high morphological plasticity. For example, it is possible to create a two-headed animal from a normal one by perturbing the physiological circuit that normally determines anterior-posterior polarity [30, 29]. This stability of a radically new target morphology (2 head shape, which continues to recur upon future rounds of amputation with no further experimental treatments) does not require genomic modifications. The old one-headed and the new two-headed animals have the same genome. Moreover the new head can be grown anywhere on the body. If one of the two heads or both of them are removed, they regenerate in the same way as they were before. Thus, two-headed planarian regenerates in a two-headed planarian with the same head locations.
It is interesting to compare how different models can regenerate a two-headed planarian. Denote by D 1 the cell structure which corresponds to one-headed planarian and by D 2 the two-headed one. After head amputation in the former, the corresponding domain is denoted by D. It is possible to amputate both heads of the two-headed animal reducing domain D 2 to the same domain D. Hence we have the same cell structure D from which we should be able to regenerate either one-headed or two-headed planarian.
Suppose that regeneration is based on emergence of patterns. Then there are two signals u and v which remain in the domain D after head amputation. Their interaction can produce some nonhomogeneous distribution of these substances by the mechanism of Turing structures. Since the characteristic diffusion time is much less (hours) than time of regeneration (days), we can reasonably assume that they will converge to some stationary distributions in the domain D before it begins to grow due to regeneration. If this stationary distribution is unique, then only a unique structure can regenerate and not two different structures with one and two heads. However, it is known that Turing structures in the same domain and for the same values of parameters can be nonunique. If there are two different structures possible in the domain D, then one of them can correspond to the one-headed animal while another one to the two-headed planarian. Convergence to these stationary solutions depends on the initial conditions. Since the distributions of u and v in D after amputation from D 1 can differ from those after amputation from D 2, then we can suppose that they will converge to two different stationary solutions. Thus, up to now, this approach admits regeneration of two different structures D 1 and D 2 from the same reduced structure D.
A limitation of this approach is that position of the second head can be arbitrary in some range of locations. If each head position should correspond to a different stationary solution in the same domain D, then we need to have possibly many such solutions or even a continuous family of such solutions. It is possible to have several different Turing structures in the same domain but if there are several dozens, then it becomes exotic from the modelling point of view and unrealistic biologically. A continuous family of such solutions is not possible even mathematically unless the domain possesses some special symmetry.
These arguments should not be considered as a proof of impossibility of this mechanism of planarian regeneration. Such complex and poorly understood processes always leave a possibility for different modelling approaches. On the other hand, the method with cell memory eagerly reproduces two-headed or multi-headed structures with any head location (Fig. 20).
Morphogenesis. The model of regeneration discussed above is based on the assumption that cells can register a signal coming from other cells. When a part of the tissue is amputated, the signal is changed. The process of regeneration consists in restoring the tissue which has the same distribution of signals.
A similar approach can be used to describe initial tissue growth. Consider a domain filled by cells. Suppose that each cell has some value ui* prescribed to it. This hypothetical signal appears in the process of embryogenesis due to some genetic and epigenetic factors. It can correspond to distribution of some morphogenes. This initial domain can be considered as an organizing center. The signal distributed inside it stimulates appearance of new cells around it. These new cells will be placed in such a way, that the signal which they produce coincide with ui* inside the organizing center.
An illustration of this mechanism is shown in Fig. 21. The initial domain is a square at the center. Depending on the distribution of the signal inside this domain, different structures can grow from it.
Morphogenesis versus regeneration. The speed of animal regeneration can be greater than the speed of its natural growth. If a 7-day old tadpoles tail is cut and allowed to regenerate for 7 more days, then new tail produced will be appropriate in size for a 14-day sized animal. It shows that remaining cells remember the size of the animal before amputation. Moreover, this is also an argument against the pattern formation mechanism. If growth is completely determined by the actual cell structure, then natural growth and growth after amputation would be the same if they start from the same cell configuration.
Target morphology. The concept of target morphology is suggested in [5, 16, 17]. The model developed in this work interprets this idea in terms of signals. Let us recall the main assumption of the model. Control cells keep the information about the old signals. At the same time they receive new signals from the growing tissue. Target morphology is the morphology for which the distribution of new signals coincides with the distribution of old signals. In the case of morphogenesis, instead of old signals, there is some given distribution which appears during embryonic development. As before, target morphology is the morphology for which the distribution of received signals coincides with the given distribution. Our conjecture is that generically the target morphology is unique, that is there exists a unique solution of the problem of minimization of signal difference.
Further developments. If regeneration is based on cell memory in the remaining tissue, then the process of regeneration can be modified if this memory is modified. Future work in our group will test this model using reagents known to modify cellular memories in the nervous system, applied to regeneration assays.
There are many possible developments of the model. Among them introduction of different cell types (differentiation) and the model of long distance regeneration. Both of them will require introduction of additional signals. At further development of this model, we are considering adding biological cells as distributed objects (not as mathematical points) and associating signals with the points at the surface of their exterior membrane, to more closely tie the modeling dynamics with known biophysical or molecular components of cellular function. Altogether, development of models such as this one, which show unambiguously (algorithmically) how shape is preserved and repaired, are an essential counterpart to genetic and biophysical data. Together, the integration of constructivist modeling and molecular biology will enable biomedical control of not only gene expression but of large-scale anatomical outcomes.