Supplementary materials
Model details not described in materials and methods
Here we give additional details about the model that were not included in the materials and methods section.
Selection of the damaged cells
The probability of cell death is dependent linearly on the damage class and equals
(A1) for ,
where is the number of damage classes equal to 20 in the presented results, and. If the probability calculated with equation A1 exceeds one or , PE is set to 1 (bold line in Fig. 3B). The parameter, equal to 0.75 throughout the paper, is the accuracy of a damage-dependent cell death, and it modulates the selectivity of the process. When then the slope of the function is proportional to and the intercept equals 0, but when then the slope equals 0 and the intercept is proportional to .
Probability of stem cell divisions
The probability of a stem cell division q is a function of the mean amount of damage in the body () and the proportion of the stem cells (). The function q reads as follows:
(A2) ,
where for or otherwise. The parameter=1/3 is the maximum probability,set to -0.05 is a shape parameter and =0.8 (or 0.3) is a boundary proportion of stem cells that determines a threshold point for the proportion of stem cells: below q drops linearly to zero. The example of the function is plotted in Figure 3.
Damage accumulation: calculation of equivalent b
The probability of changing a damage class by a cell (eq. 2 in Material and methods) is determined by two parameters a and b. In the last class this probability translates into probability of death as we assume that the amount of damage represented by last class is a limit that a cell can be still alive. We assume that this probability equals z (0.9 in our case), then the parameter b can be calculated according to the formula:
(A3) .
Probability of cell differentiation and cell dynamic
In order to keep cell proportions constant, an individual controls the numbers of stem cells and differentiated cells by means of a flexible probability of cell differentiation (differentiation factor p, see Fig. 1). This differentiation factor is calculated on the basis of recent history of an individual. To explain the process of control, we start with the formalization of the cell dynamics in one iteration (see also Fig. 1):
(A4)
where and are the numbers of stem cells and differentiated cells in the next iteration, , , and are probabilities of death for dividing stem cells, non-dividing stem cells, and differentiated cell; respectively, in the unit of time t. To keep the constant cell proportion, the polyp must fulfill the assumption:
(A5)
where is assumed equilibrium/optimal (and also initial) proportion of the stem cells. Generally, this proportion is kept constant by modulating the p value; because p is the probability and must take values between zero and one, an equilibrium proportion of stem cells and differentiated cells is not always achieved. The significant deviations from this equilibrium are observed only at a very low number of cells; i.e., when an individual is close to death. In the “Hydra in the lab” model we consider two values(for ) and (for). These values give the highest offspring production and the longest lifespan in the first case and the longest lifespan in the second case. In the “Hydra in the field” the results for a whole range of are shown and the optimal proportions of stem cells are found for both the maximum lifespan and the maximum expected offspring production.
Using the equations (A4) and (A5) we can derive formula for pt+1 for each time unit t:
(A6) .
Such calculated pt+1 is used as a differentiation probability in the next generation. In the first iteration we have no information about p, and thus it is set to an arbitrary value. This initial value has no effect on the results, as p almost immediately converges to the proper optimal value.
Introducing extrinsic mortality into the “Hydra in the field” model
The calculation of the life history of this type of Hydra model requires demographic information for one clone “in the lab.” These demographic records include information about the number of buds produced(t) in a time unit t, and the age at deaths Ci caused by internal factors for every individual i. To better illustrate the concept we can imagine that each individual i represents a clone of identical individuals and then the expected offspring production for the clone, i can be calculated according to the formula:
(A7)
whereis assumed extrinsic hazard rate.
The mean survivorship of a clone (Hydra individuals simulated for the same set of parameters) is defined as a product of two probabilities: the probability of surviving to age t caused by (i) intrinsic factors and (ii) expected extrinsic ones:
(A8)
where is the size of the considered clone in time t under intrinsically conditioned mortality only. The mean survivorship can then be used to calculate the expected lifespan for a whole clone:
(A9)
The role of parameters of minor importance
In the presented results, we manipulated only the most important parameters. The other parameters which are not presented but are potentially important are: damage rate (a), repair rate (B), size of the bud, damage selectivity (), normalization, and shape parameters(q0, q1) of the function describing the probability of division. Here we briefly show how they can affect the results. Damage selectivity is an important factor determining how precise the selection of damaged cells is. For the lower values of (see eq. A1) we observe a decreased lifespan and lifetime bud production, and an elevated mean damage class. Similar results are observed for higher damage rates (a) or lower in-cell repair rates B. Among the other factors that can affect the removal of damaged cells are the normalization and shape parameters of the function determining the probability of cell divisions (eq. A2). While the normalization parameter is used only to determine the time scale and is of minor importance, the shape parameter seems to qualitatively affect the results. Under our assumptions, the shape is concave, and when it is changed to convex the cell divisions quickly decrease with the level of accumulated damage. As a result, we get a shorter lifespan and diminished reproduction, but also a lower mean level of damage in the Hydra cells. We obtained the same qualitative results for the probability of sloughing, which is the parameter that drives the selection of differentiated cells.