On a model of pattern regeneration based on cell memory

We present here a new model of the cellular dynamics that enable regeneration of complex biological morphologies. Biological cell structures are considered as an ensemble of mathematical points on the plane. Each cell produces a signal which propagates in space and is received by other cells. The total signal received by each cell forms a signal distribution defined on the cell structure. This distribution characterizes the geometry of the cell structure. If a part of this structure is removed, the remaining cells have two signals. They keep the value of the signal which they had before the amputation (memory), and they receive a new signal produced after the amputation. Regeneration of the cell structure is stimulated by the difference between the old and the new signals. It is stopped when the two signals coincide. The algorithm of regeneration contains certain rules which are essential for its functioning, being the first quantitative model of cellular memory that implements regeneration of complex patterns to a specific target morphology. Correct regeneration depends on the form and the size of the cell structure, as well as on some parameters of regeneration.

Numerous species are able to restore complex body organs after amputation [1]. For example, planaria can regenerate their entire body from a small fragment [2], and axolotls can restore limbs, spinal cord, jaws eyes, hearts, and portions of their brain [3]. Learning to control this process is the key to transformative applications in biomedicine [4, 5], as well as pattern control of embryogenesis (with applications to birth defects). While the field is rapidly accumulating high-resolution data on the genetic networks and molecular components necessary for this process [6], fundamental insight into complex shape homeostasis is lacking. This is related to a dearth of testable models explaining the signaling dynamics sufficient to explain how the correct pattern is regenerated and how growth ceases when the proper anatomy is restored. One of the main open questions is whether the regenerating organism uses only the information available at each particular moment of time or whether it can also use information about its former state—a pattern memory [7, 8]. In the first case, in order to reproduce complex forms and different organs, we need to deal with pattern formation and emergence of forms. This is often modeled via Turing structures and other mechanisms of pattern formation and self-organisation [9, 10, 11, 12, 13, 14, 15]. The process of regeneration then depends on the regeneration of patterns. If this pattern corresponds to a stationary solution of a reaction-diffusion model, then amputation can be considered as a perturbation of this stationary solution and regeneration corresponds to decay and disappearance of this perturbation.

In contrast, the organism may keep information about its original state and restores pattern after damage by minimizing the difference between the current state and the original state (a target morphology model) [5, 16, 17]. At this time, no quantitative model of target morphology during pattern formation exists.

In this work we suggest a mathematical model based on the assumption that regeneration uses the memory of the organism about its original state. This model provides a proof-of-principle of a mechanistic model that implements patterning towards an encoded target morphology memory, and illustrates a system to formulate the assumptions necessary for regeneration of cellular structures in mathematical models.

The scheme presented below is based on the assumption that there are cells that can preserve state information for some time (memory cells). This is realistic since a wide variety of somatic cell types, not only neurons, have been shown to exhibit memory [18, 19, 20, 21, 22, 23]. We model it as follows. Suppose that a cell receives some signal u with a given intensity u = u* (this could be concentration of a signaling molecule, or a bioelectric signal [24, 25], or any other kind). After some time, when the signal disappears or changes its value, the information about the old value u* is preserved in the cell. Moreover, we presume that the cell can measure the difference between the old value and the current (new) value, u*−u, and produce another signal z with a rate proportional to this difference.

There are various kinds of cells that exhibit memory (neural cells, lymphocytes, plants, bacteria) via different mechanisms. These assumptions thus do not contradict available biological information but it is not yet known whether memory processes operate during tissue regeneration.

We suggest a possible mechanism which can provide these properties. Let the signal u correspond to the concentration of some substance A in a volume bounded by a membrane. Its value in this volume equals u* and is kept fixed. At the other side of the membrane u = 0. Its flux through the membrane is proportional to the difference of the values, that is to u*. This flux leads to the appearance of stable objects B (e.g., ion channels, molecules) which participate in processing of A (transport, consumption, reaction). The quantity of B is proportional to the flux of A, that is to u*. If u decreases, then some part of B is liberated from processing of A and becomes involved in other reactions. They lead to production of another signal z. This mechanism keeps memory either about the maximal value = u* or about the value which is kept during some time sufficient to create B.

We note that the mechanisms which allow cells to determine and to preserve the maximal signal were suggested and discussed by [13].

The signals can be transported either from cell to cell proportionally to the difference of concentrations or by diffusion through the extracellular matrix.

The implementation of the model is based on the method of cellular automata where cells are located at the nodes of a square grid. This method is a particular case of more general cellular nonlinear networks. Cells communicate with each other due to the exchange of signals and they can divide according to some algorithm described in the next section. Cellular automata are widely used to model growth of biological tissues (see [28], and the references therein). Tissue regeneration is studied for example in a more general Neuronal Organism Evolution model [26, 27]. The main objective of this work is to develop a minimal model of regeneration based on plausible biological assumptions, which would constructively explain the exact regeneration of an arbitrary (within limit) amputated part of the organism.

Consider a 2D domain D filled by cells. Each cell produces a signal u which spreads in space. Its intensity decays with distance as some function f(d). If the distance between cell i (C i) and cell j (C j) is d ij, then C j receives signal f(d ij) from C i. We will assume here that each cell produces the same signal. Therefore C i receives from C j a signal of the same intensity f(d ij). For each cell C i we can count the total signal which it receives from other cells ui=∑j≠if(dij).(1)

We will use also the notation u(x) assuming that x belong to the domain D, u i = u(x i), where x i is the coordinate of the ith cell.

Clearly, cells located in different parts of the domain will receive different signals. For example, a cell located at the boundary receives signals only from one side and the value of the signal can be less than for a cell inside the domain. Therefore the distribution u(x) represents some information about the geometry of the domain.

Let us consider some examples. Signal distribution for a rectangular domain is shown in Fig. 1. Here f(d) = 1/d 2. The value u(x) at the boundary is less than inside the domain, and the value at the corners is less than in other points of the boundary. The signal distribution depends on the function f(d) (Fig. 2).

Suppose that a part of cells is removed (white cells in Fig. 3). The remaining cells will be called control cells. We can define two signals in control cells: the old signal ui*=∑j≠i,j∈I0f(dij),i∈Ic, where I 0 is the set of cells in the original cell structure, I c is the set of control cells, and the new signal ui=∑j≠i,j∈Icf(dij),i∈Ic.

The old signal is measured in control cells from all cells in the original structure before amputation. The new signal is measured also in control cells which they receive from remaining (control) cells after amputation. These two signal distributions are different, and that difference is clearly seen near the cut (Fig. 4).

When a part of the organism is removed, the signal distribution in the remaining cells, which we call the control cells, is changed. The the question is then how to restore the cell configuration with the initial signal distribution. Remaining cells near the cut will start to divide. New cells will contribute to the signal distribution and will continue to divide filling available places in their neighborhood. This growing structure is regulated by the signal distribution in control cells and will converge to the initial structure.

We will begin with some simple examples that admit analytical solutions. Consider three cells, A, B and C. Let us remove the cell C and discuss the algorithm of its restoration. Denote by d AB, d BC and d AC the distances between the corresponding cells. Before removal the cell C, the signals received by cells A and B are as follows: uA*=f(dAB)+f(dAC),uB*=f(dAB)+f(dBC).

We put cell C in an arbitrary place and denote by d~BC and d~AC its distances to the cells A and B. The new signals received by these cells are now uA=f(dAB)+f(d~AC),uB=f(dAB)+f(d~BC).

We need to choose the position of cell C in such a way that uA=uA*, uB=uB*. These equalities are satisfied if f(d~AC)=f(dAC),f(d~BC)=f(dBC).

Since f(d) is a monotonically decreasing function, we conclude that d~AC=dAC, d~BC=dBC. These equalities determine two circles on the plane. One of them is around point A with radius d AC, another one is around point B with radius d BC. They intersect in two points. One of them corresponds to the initial position of cell C, and another one is symmetric with respect to the line AB.

Thus, in the case of three cells, one removed cell can be restored by a simple algorithm described above. A similar approach is applicable for any initial number of cells n if only one cell is removed. It should be noted that in general n−1 circles may not have a point of intersection. However in the problem considered here their intersection is guaranteed by the initial configuration.

If more than one cells is removed, the problem of their restoration does not have a simple analytical solution except for the case where all cells are located at the same straight line. The existence of solutions of the restoration problem is provided by the initial configuration. So the question is how to converge to this configuration adding new cells one after another. A possible algorithm is suggested in the next section.

We will use the difference between the signals ui* and u i, i ∈ I c in order to restore the initial form. The issue here is the algorithm of decision-making that determines where to put new cells. We will add new cells one after another and denote the set of cells in the process of regeneration by I(t). Here t is discrete time, t = t n, where at each next time step we add one cell. Let us introduce the signal ui(t)=∑j≠i,j∈I(t)f(dij),i∈Ic.

This means that we measure the total signal from old and new cells in the control cells. The purpose is to restore a structure for which ui(T)=ui*,i∈Ic for some time t = T.

We suggest an algorithm for the placement of new cells determined by the following three conditions:

1. All cells are placed in the nodes of the square grid. Each cell has 8 neighbors, 4 of them with a common side and 4 other cell with a common diagonal. Each new cell is placed in such a way that among its neighbors there is at least one cells from the cut (blastema) or another new cell. This condition provides continuity of growth of the regenerating domain, beginning from the location of the cut.

2. When we add a new cell we recalculate the signal in every control cell. The new signal should be less than or equal to the old signal, ui(t)≤ui*,i∈Ic.

In numerical simulations this condition should be satisfied with certain accuracy (Section 3.1).

Among all cells, which satisfy conditions 1 and 2, at each time step we choose that cell for which the difference S* − S(t) is minimal.