On a model of pattern regeneration based on cell memory
Let us illustrate this condition with the following example. Suppose that there are only two control cells A and B, and we choose where to place a new cell C. For each possible position of cell C, we measure the signal S(AC) received by cell A from cell C and the signal S(BC) received by cell B from cell C. We put cell C in the place where the sum of these two signals is maximal.
3b. Each control cell produces a signal proportional to the difference ui*−ui(t). This signal spreads in space and stimulates appearance of new cells. We choose the cell, which satisfies conditions 1 and 2, and where the value zk=∑i∈Icf(dik)(ui*-ui(t)) is maximal. Here k belong to the set of cells, which satisfy conditions 1 and 2.
Conditions 3a and 3b give close results. In the first case, we choose the cell which adds the greatest signal to the control cells. In the second case, we choose the cell which gets the greatest signal from the control cells. The second algorithm is more biologically plausible.
The example shown in Fig. 5 illustrates how the algorithm of cell choice works. According to condition 1, we consider all cells at the nodes of the grid near the cut (red cells). They are marked with black dots in the figure. Next, we verify that they satisfy condition 2. Some of them may not satisfy it. Fig. 5 (right) contains two additional candidates in comparison with Fig. 5 (left). This small difference appears to be crucial. We will see that condition 2 is necessary for normal regeneration. Finally, among all candidates, which satisfy conditions 1 and 2, we choose the first new cell according to condition 3a or 3b.
Continuation of regeneration without condition 2 is shown in Fig. 6. When the first row is filled, the best cell according to condition 3a is in the middle of the second row. However, condition 3b favors a cell at the continuation of the cut row (the last row of remaining cells). The algorithm with condition 3a (without condition 2) shows a better regeneration. New cells fill 5 rows of the original form and then it adds a cell outside the original form. Yellow cells show the places where condition 2 is not satisfied. The blue is the worst among such cells, that is where the difference ui−ui* is maximal.
If we use condition 2, then both algorithms (3a and 3b) correctly regenerate the original form (Fig. 7). Figs. 8 and 9 show the evolution of the new signal u i(t) in the control cells during regeneration. It gradually grows and after the regeneration of several dozens of cells it comes close to the old signal.
Let us note that the algorithm of regeneration may not preserve the symmetry of the structure. Indeed, if there are two candidates for new cells which are equivalent with respect to conditions 1–3, we first add one of them. After that the second one may not satisfy these conditions any more.
The signal used above, in order to model regeneration and morphogenesis, corresponds to chemical concentrations or to electric potential. Both of them can be described by the equation Δu-bu=h(x), where Δu is the Laplace (or diffusion) operator, the second term in the left-hand side of this equation describes consumption or destruction of this signal, h(x) is a source term. In this case the solution u(x) decays exponentially for b > 0 and as a logarithm for b = 0.
In the modelling above, we also considered polynomial decay of the solution. In order to obtain it as a solution of diffusion equation, we need to introduce nonlinear diffusion. Consider the corresponding equation (a(u)u′)′-bu=0 in the half-axis x > 0 with the boundary condition u(0) = 1. We look for the solution decaying at infinity. We can reduce this equation to the system of the first-order equations: a(u)u′=p,p′=bu.
Taking into account the boundary condition p(0) = 0, we can integrate the last equation: p2(u)=2b∫0ua(v)vdv.
From this equation and the boundary condition we obtain u(x)=1(1+cx)2/k,c=k22b2-k.
Hence, a solution exists for any k, 0 < k < 2. It can decay with the rate 1/x n with any n > 1. It corresponds to the decay rate considered in numerical simulations.
The model contains only one physical parameter, the rate of decay of the signal, and one numerical parameter related to verification of condition 2. We consider the function f(d), which shows how the signal depends on distance, in two different forms: f1(d)=d-n,f2(d)=e-nd, where n > 0. All results shown in the previous section are obtained with the first function for n = 2.
Regeneration depends on the value of parameter n and on the size of the domain. In the case of polynomial function f 1(d) = d −n, the best value of this parameter in a relatively narrow range around n = 2. If n is outside this range, then the size of well regenerated domain decreases.
Let us note that condition 2 is verified with certain precision. We require that (ui−ui*)/ui*<ϵ, where ϵ is a small positive number (ϵ = 10−14 in the simulations shown in Figs. 10–12. If this inequality is not satisfied in any of the control points (cells), then the simulation is stopped. Yellow and blue points show where the condition ui<ui* is not satisfied. Such points can appear during the simulation.
Up to now we considered a model example with a fixed shape of initial and reduced domains. The algorithm presented above allowed us to obtain the exact solution of the problem of regeneration for some class of original and reduced domains. Some examples are shown in Figs. 13 and 14 for rectangular and ellipsoid domains. We can cut off different parts of these domains and the system restores them to the original form.
Let us note that in all examples of correct regeneration, the regenerated part of the domain is convex or it is composed of several separated convex subdomains for which regeneration occurs almost independently. The algorithm described above is not aimed for nonconvex domains because the distance between two cells is measured along the straight interval connecting them. If the domain is not convex and the signal propagates along the tissue, then the distance between two points should also be measured along the tissue.
The regenerated domains presented in the previous sections are not necessarily convex. However, each regenerated part is convex. The signals produced and received by cells decay with distance, which is measured along the straight line connecting cells. In the case where the domain is not convex, the signal can propagate outside of the domain filled by cells. Biologically, this means that there are some other tissues which can transmit signals but which do not participate in the process of regeneration. Some examples of regeneration of nonconvex structures are shown in Fig. 15.
As discussed above, if the size of regenerating domain is too large, the control cells cannot distinguish signals correctly, and regeneration becomes abnormal. This critical size of the domain depends on its form. Moreover if the domain is not convex, it becomes more difficult for control cells to interpret signals and the size of the domain is even more important. Let us illustrate it with the example in Fig. 16. Correct regeneration is shown in the left image. We remove one more row of cells (middle image) and regeneration becomes wrong. We stop the process of regeneration when the first wrong cell appears (it is cell C in the middle image). Why does it appear here and not in the left image?
The nearest control cell to the cell C in the middle image is the cell B. Let uB* be the old signal in cell B, and u B(t 0) the new signal just before the appearance of cell C. The signal u BC produced by cell C and received by cell B adds to the total signal received by cell B. It remains smaller than the old signal, uB(t0)+uBC≤uB*.(2)
Therefore the second condition of the algorithm remains satisfied and cell C is added to the regenerated structure. After that regeneration continues incorrectly (Fig. 17).
In the regeneration of the structure shown in Fig. 16 (left), the nearest control cell to cell C is cell A. Its distance to cell C is less than the distance to cell B. Therefore, the signal which it receives from cell C is greater, u AC > u BC, and instead of inequality (2) we now have the opposite inequality uA(t0)+uAC>uA*.(3)
Therefore the second condition of the algorithm is not satisfied, cell C is not added and regeneration continues in a right way. The structure shown in Fig. 16 (right) also regenerated correctly. The difference in comparison with the structure in the middle image is that it does not contain cell D and other cells in the same row. The presence of cell D in the original structure increases the old signal uB* received by cell B. If this cell is absent, inequality (2) is not satisfied, add cell C cannot be added. This provides correct regeneration in the right image.
Thus, the size and organization of the cell structure are important for the correct functioning of the algorithm, especially for nonconvex structures where signals coming from different sides can mask the empty space inside the structure.
We suggest a model of regeneration of cell structures based on cell memory,—implemented as follows. Each cell receives signals from all other cells. Its total value is u*. The cell keeps a memory of this value. If the total signal changes and becomes u, the cell produces some substance with the rate proportional to u* − u. This substance stimulates the proliferation of new cells. As a result, cell structure grows until the new signal u becomes equal to the old signal u*. This mechanism allows cells to implement a kind of means-ends analysis—a working towards regeneration of a specific shape and a cessation of further growth when the correct morphology is achieved a major open question in the understanding of morphostasis in regeneration, remodeling, embryogenesis, and cancer suppression.
The main question is whether this method allows a correct regeneration of cell structure when a part of it is amputated. We show that under some additional conditions which provide continuity of growth and that the new signal cannot exceed the old signal, we can obtain an exact solution of the regeneration problem.
Limitations of the algorithm. The algorithm of cell structure regeneration suggested in this work has several limitations. First is the size of the structure. Control cells cannot correctly identify missing cells at a large distance because the signal rapidly decays. However, there is a tradeoff here. If the rate of decay is low, then the boundary of the cell structure is not clearly identified. Added cells will go beyond the original structure and the regenerated shape will be incorrect. On the other hand, rapid decay of the signal also requires high accuracy in the verification of the second condition of the algorithm. If the new signal in some of control cells becomes greater than the old one on a very small value (usually, 10−7−10−15), then the algorithm is stopped. Biologically this means that control cells are very sensitive. It should be noted however that we consider a qualitative mechanism of regeneration with arbitrary values of parameters, which we cannot relate at the moment to realistic biological parameters.
Figs. 18 and 19 show regeneration of a long thin domain for different values of parameters and sizes of the domain. There are some optimal values where the algorithm is more efficient. If we consider the size of the cell structure close to its maximal admissible value, the algorithm becomes sensitive to small changes of parameters or geometry. In particular, if the remaining part of the domain after amputation is less than some critical value (Fig. 18, left), then regeneration fails.
Finally, let us recall that the distance between two cells is measured along the straight line connecting them. If the cell structure is not convex, then this interval can be partially outside of this structure. From the biological point of view, this implies existence of other cells (not explicitly included in the model) which fill the empty space and which can transmit the signal but do not participate in the process of regeneration. It is possible to modify the algorithm is such a way that the signal propagates along the structure even in the case where it is not convex. However it then becomes significantly more complex and the path between two points to measure the distance between them can be defined in different ways. This will be explored in subsequent work.
Reaction-diffusion patterns or cell memory? Reaction-diffusion systems of equations can describe emergence of patterns. The mechanism of pattern formation mostly used in morphogenesis and regeneration is based on Turing structures. In order to describe emergence of patterns, the reaction-diffusion system should contain at least two equations and should satisfy some additional conditions. They are formulated by H. Meinhardt as short range activation—long range inhibition. Emergence of patterns is based on the interaction of these two substances (Turing called them morphogens). Since these patterns can represent stable stationary solutions, then the perturbed pattern can return to its original form after some time. This is the main idea of regeneration models with reaction-diffusion equations.