Aging is agings: a recursive definition of biological aging(s); part 3, Recursion

In the first part of this study, Aging is agings: towards a recursive definition of biological aging(s); part 1, definition the following definition of biological aging(s) was introduced:

Biological aging is agings underneath, the result of multiple, separate, diverse, interconnected, but malleable processes, eventually compromising normal functions of the organism at different rates and at all levels.

In the second part Aging is agings: towards a recursive definition of biological aging(s); part 2, Explication we argued why the ‘need and must’ to come up with a consensus definition and that there’s strong reasons it should be a so called explicative definition a la Carnap.

Today we arrive at the heart of the heart of the heart our definition, recursion. šŸ™‚

5. What kind of definition is offered ? A recursive definition 

I think the most important insight of my definition proposal is that the complexity and broad-spectrum of biological aging can be captured and the internal structure of the separate aging processes can be laid out and accounted for by applying a so called recursive definition that is expressed at the start of the definition as ā€˜Biological aging is agingsā€¦ā€™.

But what are recursive definitions and how can they be formally correct? The corresponding Wikipedia entry is a good starting point, so let’s keep it simple.

In general, recursion, the procedure behind recursive definitions, always involves applying the entity we would like to work with magically to itself somehow. The magic lies in this applying X to itself but, once we take a closer look to the internal structure of this procedure, the paradoxical magic evaporates, as numerically the thing applied to itself is not one, not the same, but several. Yet, effectiveness of the procedure remains. Recursive definitions are mostly known from mathematics, so letā€™s see 2 examples there using the wikipedia entry to keep advanced stuff to a minimum.

Thereā€™s a deep connection (equivalence under some conditions) in math between induction and recursion, so recursive definitions are also called inductive definitions.

In the “An introduction to inductive definitionsā€, set theorist Peter Aczel describes recursive definitions in terms of elements of a set that are defined with the help of other elements within same set. 

Example #1 Set example: N of natural numbers

(1) 1 is in N.

(2) If an element n is in N then n + 1 is in N.

(3) N is the intersection of all sets satisfying (1) and (2)

A very important concept is the base case and in the example above 1 is playing the role of the fundamental unit, out which the other elements of natural numbers can be built up, one by one.

But elements of recursive definitions might be different mathematical objects or structures than just sets, for example they can be functions.

Example #2: Function example: factorial function n!

(1) 0! = 1.

(2) (n + 1)! = (n + 1)Ā·n!.

You can see how f(n+1) is defined via f(n), showing the crucial step in induction/recursion.

The nemesis of correct and successful recursive definitions is infinite regress, this happens when the fundamental building block, the base case is used to be defined with itself, so does not escape the definition and let circularity set in. Funnily enough, Peter Aczel, mentioned above is the founding figure of the so called non-standard set theory that allow sets, hypersets, to be elements of themselves. This non-wellfounded set theory abandons the Foundation Axiom and somehow still manages to build an exciting new theory out of an apparently circular base structure.  

In general, circular definitions are well known to cause trouble in linguistic environments, leading to well-known paradoxes, but somehow math has a way of managing them.

One explanation might be that in math, the examples are more of an object circularity kind, like sets being elements of themselves, while in other ares, like language, they are more of a definition circularly kind, like. These circularities generally indicate trouble, but this is not necessarily so.

In any case, if one suggests a consensus and formally correct definition of biological aging, it is a jolly good idea to propose one, that does not involve infinite regress.

So non-circular, recursive definitions have a base case that is the

  • definiens for other elements,
  • satisfies definition without being defined by it,
  • thereby terminates recursion, stops infinite regress.

Now switching back to the Hallmarks framework, please see Figure 6 from paper below and recall the recursive first part of the proposed definition:

ā€˜Biological aging is agings underneath, the result of multiple, diverse, separate but malleable processesā€¦ā€™

How can recursion elaborate further on the hallmarks framework and provide an internal structure to the processes proposed? Here we need to cite at length to how the authors of the Hallmark framework summarise the 3 types of hallmarks they propose and the connections between them:

‘A global view of the nine candidate hallmarks of aging enumerated in this Review suggests three categories: primary hall- marks, antagonistic hallmarks, and integrative hallmarks (Figure 6). The common characteristic of the primary hallmarks is the fact that they are all unequivocally negative. This is the case with DNA damage, including chromosomal aneuploidies; mitochondrial DNA mutations; and telomere loss, epigenetic drift, and defective proteostasis. In contrast to the primary hallmarks, antagonistic hallmarks have opposite effects depending on their intensity. At low levels, they mediate beneficial effects, but at high levels, they become deleterious. This is the case for senescence, which protects the organism from cancer but which, in excess, can promote aging. Similarly, ROS mediate cell signaling and survival but, at chronic high levels, can pro- duce cellular damage; likewise, optimal nutrient sensing and anabolism are obviously important for survival but, in excess and during time, can become pathological. These hallmarks can be viewed as being designed for protecting the organism from damage or from nutrient scarcity. But when they are exacerbated or chronic, they subvert their purpose and generate further damage. A third category comprises the integrative hallmarksā€”stem cell exhaustion and altered intercellular communicationā€”that directly affect tissue homeostasis and function. Notwithstanding the interconnectedness between all hallmarks, we propose some degree of hierarchical relation between them (Figure 6). The primary hallmarks could be the initiating triggers whose damaging consequences progressively accumulate with time. The antagonistic hallmarks, being in principle beneficial, become progressively negative in a process that is partly promoted or accelerated by the primary hallmarks. Finally, the integrative hallmarks arise when the accumulated damage caused by the primary and antagonistic hallmarks cannot be compensated by tissue homeostatic mechanisms. Because the hallmarks co-occur during aging and are interconnected, understanding their exact causal network is an exciting challenge for future work.

The Hallmarks of Aging

According to this proposal, the 4 primary hallmarks form the undisputed base cases of agings, these being the processes that arise by themselves due to the normal trajectory fo human lives and need no further, lower-level building blocks to account for them within the definition. So they not need be defined by the definition itself.

But the big question is whether the second type, the so called antagonistic hallmarks, can be considered further base cases or derivative complex processes. These three processes are metabolic changes (deregulated nutrient sensing), mitochondrial dysfunctions and cellular senescence. Depending on their intensity, they have opposite effects.

On the one hand, the antagonistic hallmarks do arise by themselves with age, which suggest they might qualify as base cases, next to the primary hallmarks.

On the other hand, these processes supposedly turn into negative processes from a theoretical proposed positive lowly expressed state partly due to the triggers of the primary hallmarks.

The term antagonistic suggests that something funny is going on with these processes and looks like they are indeed ambiguous, and look semi-autonomously formed.

If we look for the first aspect mentioned above: autonomy, they are base cases, if we look for the second ā€˜semiā€™ aspect, they are complex cases.

How can we resolve this situation within a recursive definition? 

First, please note something else and this is comes from science: 2 out of these 3 antagonistic processes are the ones where modulating these processes with interventions yielded the biggest benefits in terms of healthy lifespan. Think about calorie restriction related to nutrient sensing (rapamycin, NAD boosters, ā€¦) and think about senotherapetuci agentsā€¦.

If the definition wants to be actionable, then it is quite important to accommodate these processes into it in a satisfiable manner.

Since interventions suggests that these processes are quite crucial to the overall phenotype, and are in a position to trigger, to elicit acceleration/deceleration of biological aging, my current solution is to treat them as bases cases, that can be alone sufficient enough to be qualified/quantified as biological aging. 

What we lose here in the definition is the connection between the primary hallmarks and these antagonistic hallmarks, so the second aspect mentioned. It becomes implicit in the current phrasing of the definition. This way, some part of the internal structure and connectedness of these hallmark processes seem to be lost. But, we can make one amendment here, we can add the term ā€˜interconnectedā€™ in our listings of the features of these processes of biological aging.

So instead of: ā€˜Biological aging is agings underneath, the result of multiple, separate, diverse, but malleable processesā€¦ā€™

we write:

ā€˜Biological aging is agings underneath, the result of multiple, separate, diverse, interconnected, but malleable processes,ā€™

This explicit acknowledging of the links between the separate processes might come handy later on and can be generalised further. And it actually very much in sync with the recent trends in research showing more and more links between these processes.

Concerning the integrative hallmarks, these ā€˜arise when the accumulated damage caused by the primary and antagonistic hallmarks cannot be compensated by tissue homeostatic mechanismsā€™. Here we have a clear statement of the processes triggering these hallmarks, including exhaustion of regenerative stem cell reservoirs across the body and inflammaging/immunosenescence, namely the primary and the antagonistic. These integrative hallmarks can be built on top of the previous hallmarks then in the definition. This way, weā€™ve gotten just one more argument on why the antagonistic hallmarks should be considered base cases and not derivative processes, since they are the triggers, ā€˜building blocksā€™ of the integrative hallmarks.

Now we are in a position to resolve two general problems affecting recursive definitions concerning our proposed definition in the next section.

6. Resolving two problems with recursive definition: multiple base cases and circularity 

Problem #1: Is it ok to have more than one base cases in a recursive definition as it is proposed by this definition?

Answer: Absolutely, consider how the Fibonacci series is being constructed, 2 base cases are needed: n=0 and n=1 in order to construct the third element of the series as being the sum if the previous two.

But there might be recursive definitions where it is central to allow 3, 4 or n number of finite elements. Letā€™s not discuss the infinite base case here. šŸ™‚

In our proposed definitions of biological aging, currently we propose 6 base cases, but this number itself is not central or necessary, and it can be more. what matters is that we have a relatively small number of cases that seem to exhaust biological aging, or describe near-completely. More on that later. 

Problem #2: Does the current definition exclude infinite regress? 

Answer: Yes, the base cases do not refer to themselves and are primitives. Here again, characterising antagonistic hallmarks as base cases is important as we avoid those being constructed with the help of other base cases, the primary hallmarks. The composite cases, the integrative hallmarks do refer to base cases, but do not refer to themselves. Please note that technically composite elements can be made out of other composite cases, but we are not using this construction here.

Also keep in mind, that avoiding circularity at all cost is not necessary here, as it might actually work in some definitions already, see math example above, yet it is certainly highly recommended. Who knows maybe there will be proposals later arguing for allowing circularity in the definition of biological aging.

We have finished here explaining how recursion in our recursive definition of biological aging is constructed and how it avoids some problems these definitions might be especially prone to. In the next post we ask how does this recursive definition enable flexibility, scalability and actionability, so all the good stuff?