06 December 2016

Is it possible to program aging (5)

Spatial models

Continuation. The beginning of the article is here.

Travis (Travis (2004))

Some of the ideas supposedly explaining programmed aging include concepts of kin selection, often taking into account the spatial dimension. A detailed spatial model was described by Travis (2004), who developed his theory using agent-oriented computer simulations. In this model, individuals adhere to the rules of reproduction and death indicated in Table 2. Agents can die from age-independent mortality "e" or die (reflecting programmed aging) upon reaching terminal age "d". Reproduction ensures that the voids of the habitat niche are filled by descendants with a probability decreasing with age a and is indicated as ca—1. Travis used an evolutionary approach to find the optimal value of the age of programmed death "d", similar to the approach that the authors applied in the model for the idea of Goldsmith (2008). Simulations have demonstrated that "d" does either increase or decrease to reach a certain age of death. Travis came to the conclusion that an individual can increase its initial fitness through death if the niche of the neighborhood being vacated is filled with a newborn with higher fertility. The authors fully agree with this explanation, but the critical point is that it is relevant only if fertility fades with age. In his analysis, the author himself points out this: if the reproductive ability does not fade with age, programmed aging does not evolve. The model does not explain the evolution of physiological aging from an ageless state. What, then, is the essence of the idea? Obviously, the theory that aging is genetically programmed should explain how such a program can arise from an ageless state.


Table 2. A set of rules describing the behavior of agents according to Travis (2004). "A" corresponds to the current agent, and "E" corresponds to an empty niche. The subscript "N" corresponds to a field from Moore's neighborhood, the absence of an index indicates the agent's current field.

Mitteldorf and Pepper (Mitteldorf & Pepper (2009))

Another model based on the territorial proximity of agents was proposed by Mitteldorf and Pepper (2009). In contrast to the Travis model (2004), in this model fertility is maintained at a constant level and describes the probability that an agent will choose a random niche from his neighborhood for reproduction. If it is empty, a descendant appears, otherwise reproduction does not occur. Also in this model, agents can die from age-dependent background mortality c, or die upon reaching the programmed maximum age of maxLifespan. However, in this case, there is a small additional probability of epiProb that the agent generates an epidemic that simultaneously kills all individuals of the group to which it belongs (Table 3). If the evolution of maxLifespan is allowed, as described earlier, an optimal maxLifespan is formed, depending on other parameters of the model. Thus, the authors suggest that physiological aging is an adaptation that limits the spread of diseases.


Table 3. A set of rules describing the behavior of agents in the Mitteldorff and Pepper model (2009). "A" corresponds to the current agent, "E" corresponds to an empty niche, and "X" corresponds to any content of the niche. The subscript "N" corresponds to a field from Moore's neighborhood, the subscript "P" indicates agents belonging to the same group, and the absence of an index indicates the current field of the agent. The displacement rule is not part of the Mitteldorff model, however, it was added by the authors to explain the operation of the original model.

The authors re-applied this model using the MASON library and tried to reproduce Figure 2 by Mitteldorff and Pepper (2009), demonstrating the dependence of the formed maxLifespan on background age-dependent mortality. However, for certain substitutions of parameters, the population dies out after a short period of time (Figure S2).


The authors suggested that in the original figure there may be an error in the emerging maxLifespan (or fertility), since in other diagrams of the authors maxLifespan is lower by an order of magnitude. Therefore, for their simulations, the authors reduced fertility to 0.1, which allowed them to reproduce the simulation results that are quantitatively identical to the Mitteldorff results (Figure 4, A).

This means that in an environment with low background mortality, agents die at a young age as a result of programmed aging, whereas in an environment with high risks, programmed aging occurs at very late stages of life. This result is unexpected and illogical, since usually a high risk from the environment is associated with a high rate of aging, whereas long-lived individuals are often found in a safe environment. And indeed, there is a huge amount of experimental evidence for this correlation, as has been shown in field studies on possums and daphnia, in experimental evolutionary studies on fruit flies and in studies analyzing data on the survival of many species of birds and mammals, as well as poisonous and non-poisonous animals. At the same time, there is only one study on guppies, the results of which do not correspond to this trend. The developers of the model are familiar with these data, but do not consider them as a counterargument.


Figure 4. (A) Emerging values of maximum life expectancy as a function of background mortality in the simulation of the Mitteldorff and Pepper model (2009). The higher the background mortality, the higher the optimal value of the programmed age at which agents die. (C) Background mortality and programmed death upon reaching maxLifespan are two options for dividing a group of agents into two smaller groups, which weakens the effects of the epidemic.
However, a similar effect can be achieved in cases where agents have the ability to move (shaded agent).

A more detailed analysis of the rules shows why background mortality and programmed death in this model correlate in reverse order. Both processes are ways of dividing large groups into smaller ones, which limits the destructive effect of epidemics (Figure 4, C). Since there must be an optimal separation rate that balances positive (weakening of epidemics) and negative (destructive agent effects), high background mortality is associated with a low frequency of programmed death and vice versa. To verify this explanation, it is sufficient to establish one additional rule that provides the possibility of moving agents (Table 3). As can be seen in Figure 4B, this also leads to the division of large groups into smaller ones, but without destroying the agents. In the computer simulation shown in Figure 5, maxLifespan and moveSize can evolve simultaneously. The system was initiated with a low background mortality (c = 0.01) corresponding to the optimal maxLifespan (the left point on the graph in Figure 4A) and the initial moveSize value equal to zero. As can be seen in the figure, the system simultaneously increases the values of moveSize and maxLifespan. This means that agents whose group size is reduced due to displacement are more adapted compared to agents who are unable to move and divide their groups due to programmed death. This behavior can only be formed as a result of the fact that the original set of rules established by Mitteldorf & Pepper (2009), does not provide the ability to move.


Figure 5. Simultaneous evolution of maxLifespan and moveSize in the simulation of the modified Mitteldorff and Pepper model (2009). The simulation was initiated from the maxLifespan equilibrium point for c = 0.01 (the left point on the graph in Figure 4A). In addition, moveSize can evolve with a step length of epsMove = 0.1. In such conditions, the system develops in the direction of higher values of moveSize and maxLifespan.

Martins (2011)

Martins (2011) presented a very interesting simulation based on a simple set of rules (Table 4). In this model, reproduction can occur not only in empty nearby areas, but also in niches occupied by other agents. In this case, either a newborn or an existing agent dies with a probability depending on the fitness of both agents. Fitness is a property inherited by an agent from a parent that has deteriorated or improved by the value of "M" (due to mutations). In addition, the fitness of each agent decreases at each time step by the value "d", which presumably reflects environmental changes that make all agents slightly less fit. Agents can die not only as a result of the struggle, but also upon reaching a certain maxLifespan reflecting programmed aging. After that, Martins (2011) conducted several experiments on direct competition between aging and ageless agents and demonstrated the existence of an optimal maxLifespan value, which he considers evidence in favor of the possibility of the evolution of programmed aging.


Table 4. A set of rules describing the behavior of agents according to Martins (2011). "A" corresponds to the current agent, "X" corresponds to the contents of any field, and "E" corresponds to an empty field. The subscript "N" indicates a field in Moore's neighborhood, and the absence of an index indicates the agent's current field.

The authors also reproduced this model using the MASON library and performed stimulations in which the value of maxLifespan could evolve as described earlier for the Goldsmith model (2008). Figure 6A shows a typical simulation result confirming that the evolutionarily optimal value of maxLifespan for the stated parameters is about 5.5 (d = 0.01, M = 0.03). The authors also simulated direct competition between agents with a maximum life expectancy close to the optimal value (maxLifespan = 5) and agents with an upper limit value life expectancy of 50. Since the annual mortality of agents is approximately 50% (due to the struggle with newborns), this is equivalent to an ageless phenotype. Figure 6B shows the development of important model variables in the course of competition, leading to the extinction of ageless species. The ratio between the fitness of aging and non-aging species quickly reaches values between 1.1 and 1.2, while Martins (2011) also noted that aging agents are characterized by higher fitness. What could be the reason for such a higher fitness? The low ratio of the values of the life expectancy of generations indicates the answer. Since aging agents are characterized by a shorter generation lifespan, favorable mutations (continuously generated by "M") spread faster in such populations. Despite the existence of associated costs in the form of a reduced number of births throughout life, apparently, increased fitness outweighs this disadvantage. In general, the Martins model (2011) is a practical embodiment of the theoretical ideas of Libertini (1988).


Figure 6. (A) The formation of the maximum life expectancy during evolution in the simulation of the Martins model (2011). The age at which individuals die increases or decreases to the optimal value, which depends on the parameters of the model. (C) The temporal dynamics of the model variables during the competition experiment between agents without a maximum lifespan (maxLifespan = 50) and agents with a lifespan close to the maximum value (maxLifespan = 5). In this model, the two-dimensional world was randomly initialized by 40% of aging and 40% of ageless agents. The other parameters are similar to the parameters of the model (A), but with fixed values of maxLifespan (i.e. epsLifespan = 0).

However, the model is based on unrealistically rapidly changing conditions in combination with the constant appearance of favorable mutations. The extinction of fitness caused by "d" occurs so quickly that environmental conditions change significantly during the life of one agent. In addition, while the agent is still alive, there are so many new favorable mutations (caused by "M") that after a few time steps, the fitness of chronologically old agents is lower than the fitness of newborn agents. We can eliminate one of these problems by preventing the extinction of fitness (d = 0), while maintaining a positive value for new mutations (M > 0). In such conditions, fitness will gradually increase, whereas at d > 0 its value reaches a stable level (Figure S3).


However, this is not a problem, since the result of the struggle between agents is determined by relative fitness, which is not an absolute value. Interestingly, in direct competition between aging and ageless genotypes at d = 0 in this case, the ageless genotype wins in >95% of cases. Figure 7A shows a comparison of the fitness level of aging and ageless agents during the simulation of such competition for d = 0 and d = 0.01. In both cases, the aging phenotype has a higher level of fitness (ratio > 1), however, for d = 0 this is predominantly less pronounced than for d = 0.01. When considered together with a smaller number of births throughout life (Figure 6B), this is enough for the Libertini mechanism to stop contributing to the programmed lifespan. This is also confirmed by ensuring the possibility of maxLifespan evolution over time (Figure 7B). Instead of applying a fixed value as in Figure 6A, the age of programmed death evolves in the direction of constantly increasing values, slowing down only under the influence of the weakening of the force of natural selection as the chronological age increases.


Figure 7. (A) Temporal dynamics of the fitness ratio of aging and ageless agents during the competition experiment. For d = 0, the aging phenotype still retains an advantage in fitness (ratio > 1), but it is lower than for d = 0.01. Under these conditions, the ageless phenotype wins. (C) The development of maxLifespan during evolutionary periods for d = 0. Whereas for d > 0 maxLifespan reaches an equilibrium value (Figure 6A), this does not happen when the possibility of stable extinction of fitness is canceled.

The Martins model (2011) has another more fundamental problem. The agents of this model are carriers of two properties (genes) that evolve over time, namely, the maximum life expectancy, upon reaching which agents die, and fitness, which determines the result of competition between agents. The simulation created by Martins describes a population of clones reproducing asexually. However, in reality, all higher animals reproduce sexually, as a result of which the offspring receive the genes of both parents. Of course, in the theory of the life cycle, the advantages and disadvantages of the approach to organisms as asexual or sexually reproducing have been repeatedly considered. In particular, when using analytical models, in many cases it is much easier to work with asexual organisms, but this may cost the loss of biological realism. At the same time, it is relatively easy to expand the computational model in such a way that it extends to sexual heritability. Therefore, the authors wrote their simulation in such a way that the agents could reproduce sexually. In this case, the mating partner is selected from the environment, and the values of maxLifespan and fitness acquired by the offspring are borrowed from one of the parent individuals. After that, the authors repeated the evolution of maxLifespan, the value of which in the conditions of sexual reproduction reached the optimal level of 5.5 (Figure 6A). However, when using more realistic parameters of sexual reproduction, a completely different result is obtained (Figure 8). Instead of reaching a stable equilibrium level, the maxLifespan value increased continuously and had no limit value. The reason for this is simple. Since the genes that determine maxLifespan and fitness in this case may belong to genetically different parents, it is possible that a descendant may appear who simultaneously inherited a long lifespan and high fitness. In the short term, these offspring turn out to be more adaptable than their parents, which gives them an advantage in competition. Thus, the mixing of genes that occurs during sexual reproduction leads to the appearance of "freeloaders" who enjoy high fitness without paying for it with a short lifespan. Under such conditions, there is no evolution of programmed aging in the Martins model, even with a continuous decrease in fitness (d > 0).


Figure 8. The development of maxLifespan over evolutionary periods in the simulation of the Martins model (2011), where agents reproduce sexually and inherit the maxLifespan (L) and fitness (F) genes from both parents. When using the same parameter values as in Figure 6A, maxLifespan increases without restrictions. The inset shows that if an agent with a large maxLifespan and low fecundity (Lf) is crossed with an agent with a small maxLifespan and high fecundity (lF), the result may be the appearance of all possible combinations, including offspring with a long lifespan and high fecundity (LF).

Mitteldorf and Martins (Mitteldorf & Martins (2014))

It was mentioned above that the Martins model (2011) implies unrealistically rapidly changing environmental conditions, which leads to a loss of fitness throughout the life of individual agents. In order to avoid this problem, Mitteldorf and Martins (2014) introduced a new model with slightly modified rules. In this model, agents still inherit the fitness level from their parents, but it no longer decreases at each time stage. Moreover, in this variant of the model, it is possible to correct the ratio of favorable and unfavorable mutations affecting the fitness of descendants, so that favorable mutations (increasing fitness by +1) occur with a probability of 1/(1+D), and unfavorable mutations (reducing fitness by -1) – with a probability of D/(1+D). Thus, at D = 0, only favorable mutations will occur, at D = 1, the probability of occurrence of favorable and unfavorable mutations is the same, and at D = ∞, only unfavorable mutations occur.

The rest of the rules are similar to the rules of the earlier model in that agents can spawn offspring in both empty and occupied cells of the environment, as well as in that agents die after reaching a certain maximum lifespan. The only addition is that in this case, agents can also die from age-independent mortality "m" (Table 5).


Table 5. A set of rules describing the behavior of agents according to Mitteldorf and Martins (2014). "A" corresponds to the current agent, "X" corresponds to any contents of the field, "E" corresponds to an empty field. The subscript "N" indicates a field in Moore's neighborhood, the absence of an index indicates the agent's current field.

Figure 9A (solid lines) demonstrates that in this new model, the genetically programmed maximum life expectancy also reaches a certain value if it cannot evolve freely. In the model simulation shown, a value approximately corresponding to five is reached either from below (starting with three) or from above (starting with eight). This is unexpected, since when studying an earlier similar model (Martins, 2011), it was shown that the elimination of a continuous decrease in fitness leads to an unlimited increase in maxLifespan (Figure 7B). Interestingly, the new model differs from the older one in that the result of the struggle between the newborn and the previously existing agents is predetermined. In the old model, the probability of victory was proportional to the fitness ratio of both agents f1/(f1 + f2), whereas in the new model, the fight begins only if the fitness of the newborn is higher than the fitness of the neighbor, after which the newborn agent wins with probability "P" (the default value of which is 0.5). The reason for using this new procedure is unclear, since it is neither simpler nor more realistic than the old method. However, it is obvious that it has a dramatic effect on the result of the simulation. If the model of Mitteldorff and Martins (2014) is only slightly modified in such a way that it uses the old procedure, then the evolution of maxLifespan in this case leads to a continuous increase in ages (Figure 9A, dotted line).


Figure 9. (A) The formation of maximum life expectancy during evolutionary periods according to Mitteldorff and Martins (2014) (solid lines) and subject to the rule of superiority in the struggle between agents applied by Martins (2011) (dotted line). In the first case, a certain value of the maximum lifetime is formed, and in the second case, the maxLifespan increases without restrictions. (C) An experiment on competition between aging (maxLifespan = 5) and ageless (maxLifespan = 5000) agents in an extended version of the model, in which the probability of occurrence of a mutation that changes the fitness of the offspring can be set using the mutProb parameter. The lower the value of this probability of mutation occurrence, the less likely the aging phenotype wins in the fight against the ageless phenotypes (each column reflects 100 cases of competition).

One of the goals of creating a new model was to increase its realism by eliminating the constant decline in fitness represented by the rapidly changing environment in the original Martins model. However, the new model still implies another unrealistic assumption, namely an unrealistically high frequency of mutations. Each descendant has a mutation that changes its fitness in one direction or another. Since the total population size is approximately 60,000 individuals breeding at each time stage, approximately 30,000 positive and approximately 30,000 negative mutations occur each year, which ensures a variety of fitness genotypes in the population (Figure S4).


The parameter "D" introduced by the authors regulates the ratio between positive and negative mutations, but the overall frequency of mutations remains at the same level. Therefore, the authors expanded their model in such a way that mutations affecting fitness occur only with a certain low probability of mutProb, and conducted experiments on direct competition between aging and ageless agents for different values of the probability of mutations. As can be seen in Figure 9B, the success rate of genotypes with programmed death decreases as the mutProb value decreases. At the level of 10-6, which corresponds to about 1 mutation per 30 time stages per population (Figure S4), the aging genotype wins only in about 4% of the fight episodes. The reason is that the frequency of mutations directly affects the proportion of carriers of beneficial mutations, the spread of which can be accelerated by shortening life expectancy (according to the Libertini equation). Despite the fact that the total number of mutations per genome per generation is approximately 1 (Keightley et al., 2014), the number of positive mutations will be much less. And mutations with a pronounced positive effect will be extremely rare. Thus, to provide more realistic values of mutation frequency, the carrier fraction decreases so much that programmed aging loses the ability to evolve.

Finally, the authors also analyzed the response of this model to sexual reproduction. The mating partner is selected from the environment, and the values (genes) for fitness and maxLifespan of the offspring are randomly inherited from one of the parents. The results shown in Figure 10 are very similar to the results of the corresponding simulations for the original Martins model (Figure 8). The age of programmed death increases without an upper limit, this growth is slowed down only by a decrease in the pressure of natural selection. Since the number of surviving agents decreases as the chronological age increases, the breeding advantage of further increasing maxLifespan decreases, which manifests itself by slowing the growth of maxLifespan. Thus, under the condition of sexual reproduction, this model also does not provide the evolution of programmed aging.


Figure 10. The formation of maxLifespan during the evolutionary stages in the simulation of the Mitteldorff and Martins model (2014), where agents reproduce sexually and inherit the maxLifespan L and fitness F genes from both parents. When using the same set of parameters as in Figure 9A, maxLifespan in this case increases without an upper limit. The inset shows that if an agent with a high maxLifespan value and low fecundity Lf is crossed with an agent with a low maxLifespan value and high fecundity lF, all possible combinations can occur, including offspring with a long lifespan and high fitness

Werfel (Werfel et al. (2015))

One of the latest assumptions about the reality of programmed aging was made by Werfel and co-authors (2015), who developed a spatial simulation containing two different types of agents: resources and consumers. Resources obey only one rule, namely reproductions in free fields of the Neumann neighborhood with probability "g" (Table 6). Consumers exhibit more complex behavior and obey three rules. They multiply by converting the resource contained in the environment into a new consumer with a probability of "P", they may die due to the depletion of the environment (with a probability of "v"), leaving an empty field, or they may die as a result of programmed death with a probability of "q", leaving a resource behind. Computer simulations by Werfel (2015) showed that if "q" can evolve by increasing or decreasing its values for descendants by a small value of "epsilon", in such cases the base mortality reaches a final value exceeding zero. The authors of the article accepted this as proof of the possibility of the evolution of programmed aging in spatial systems.


Table 6. A set of rules describing the behavior of agents in the Shipyard (2015). "R" corresponds to a resource, "C" corresponds to a consumer, and "E" corresponds to an empty field. The subscript "N" indicates a field in the Neumann neighborhood, the absence of an index indicates the agent's current field. The movement rule for consumers is not part of the Werfel model, but it was added by the authors in order to explain the principle of operation of the original model.

Unfortunately, the creators of the model did not provide a clear explanation of the need for the evolution of basic mortality, that is, they did not explain the breeding advantage provided by programmed death. Therefore, the authors reproduced their model using the same MASON library. This allowed them to confirm that the baseline mortality q evolves to an optimal value depending on the model parameters (Figure S5).


However, with a detailed analysis of the set of rules for the resource and the consumer, it becomes obvious that the rule for programmed aging provides consumers with the ability to move – a property that they do not otherwise possess. Figure 11 demonstrates a sequence of five events that allows a pair of consumers (top row) to move one grid division to the right (bottom row) by temporarily creating several resource agents using the programmed aging rule. First, the consumer on the right dies and leaves behind a resource. After that, the resource produces a second resource to its right. The next stage consists of two steps in which consumers multiply by converting a nearby resource into a new consumer. And finally, the extreme left consumer dies from the exhaustion of the environment. Obviously, the ability to move is very useful for isolated consumers, as it gives them a chance to obtain resources and reproduce by converting these resources into consumers with the same genotype.


Figure 11. A sequence of events demonstrating how a 1D group of two consumers (top row) can move one grid element to the right (bottom row) in five steps (shown on the vertical axis) according to the Werfel rules (2015). The event type is indicated by a subscript for the agent performing the stage. So at the first stage, the consumer located on the right undergoes programmed death (index q), leaving a resource in the field being released.

To verify this explanation, the authors applied a new rule that allows consumers to move through the grid elements with a moveSize coefficient at each stage of the simulation (Table 6). For example, if moveSize = 1.7, there is a 70% chance of moving two steps and a 30% chance of moving 1 step.

The authors carried out a simulation in which they allowed the parameter q to evolve over 5,000 stages (starting from 0), as a result of which its value reached a stable level, approximately corresponding to 0.13. After that, they allowed the moveSize parameter to evolve and, as can be seen in Figure 12, this led to a constant increase in the value of this parameter with a simultaneous decrease in the q value to about 0.02. Thus, if possible, the system prefers a non-lethal method of movement rather than a suicidal one.


Figure 12. Evolution of the q and moveSize parameters in the simulation of the modified Werfel model (2015). At the first 5,000 stages, only the q value can evolve, reaching an equilibrium value of approximately 0.13. After that, the ability to evolve from the zero level gets the moveSize value. As a result, the moveSize value continuously increases, while the q value decreases to about 0.02. The inset shows a snapshot of the simulation zone, where resources are indicated in yellow and consumers in purple.

To test the assumption that a large moveSize value can completely prevent the appearance of a finite q value, simulations were carried out in which the moveSize value was fixed at various levels and only the q value could evolve to a stable value. Figure 13A confirms that an increase in the moveSize value leads to a decrease in stable q values, but even for exceptionally high values, q always remains above zero. Perhaps this still provides an additional selective advantage by further increasing the ability to move (both methods of movement are additive) or q ss is maintained due to the balance between mutations and selection. In this case, q > 0 is maintained in the population not because it provides a breeding advantage, but only because it is constantly reproduced due to mutations. One of the methods of verifying this statement is to change the frequency of mutations. If q ss is maintained by selection, the mutation frequency should not affect the stable value, but should depend on it if the balance between mutations and selection is maintained. Therefore, the moveSize value was maintained at 50, and the mutation rate was reduced to below 0.2. This value was used in earlier simulations. Figure 13B clearly demonstrates the dependence of q ss on the frequency of mutations and, accordingly, its maintenance due to the balance between mutations and selection.


Drawing. 13. Simulation results of the modified Werfel model (2015). (A) If the moveSize value is set to increase, q evolves to progressively lower stable values. However, even for moveSize = 50, the qss value is > 0. (B) If the probability of mutation q is reduced for the value moveSize = 50, the corresponding stable value q also decreases. This indicates that the equilibrium value of q in the population is maintained only due to the balance between mutations and selection.

In general, all this confirms the authors' suspicion that the results of Werfel (2015) can be explained by the process of related selection, in which life expectancy is sacrificed in favor of allowing genetically related genotypes to move, thus achieving the resources necessary for reproduction.

End: Discussion

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