GOMPERTZ MORTALITY, STREHLER AND MILDVAN'S THEORY, OLOVNIKOV'S TELOMERES, MUSHKAMBAROV'S MEIOSIS AND AGING

Gompertz law, Olovnikov's telomeres and disputes on negligible senescence

Here, I’m going to reminisce a bit about a highly technical paper (Tarkhov et al. 2017), and a way more interesting acquaintances it opened doors for me to. The paper attracted some attention from local gerontologists, and I was invited to give a talk at a gerontological seminar at Emanuel Institute of Biochemical Physics of Russian Academy of Sciences, which the late Alexey Olovnikov attended. He was the first to put forward the telomere hypothesis of aging in 1971, at least 11 years before Elizabeth Blackburn, Carol Greider, and Jack Szostak who received the Nobel Prize for telomeres in 2009. You can read more about this Nobel controversy here, but, rather, please, read his legendary paper on telomeres.

Later, another late Russian gerontologist Nikolai Mushkambarov reached out to me about his book Gerontology in Polemico, which he published in 2011. It turned out that he came to the same conclusion as we did about the Strehler-Mildvan correlation 6 years before us. It was a great find for me, and it is a great book, which I highly recommend to read to anyone interested in aging — though, it’s not translated into English yet. He challenged the existence of negligible senescent species in his book, and emphasised the role of meiosis in rejuvenation. He dreamt about figuring out how we could reactivate meiosis mechanisms to rejuvenate old cells. I definitely learnt a lot from him and his book, and I agree with him on most of the questions he raised in his book. Later, he also invited me to give a talk to his students at the Pirogov Russian National Research Medical University, which is considered the best medical university in Russia. It was definitely a big deal for me, a theoretical physicist, to present some abstact aging research to real future medical doctors wearing white lab coats.

Back to the sciency part. People have been applying mathematical tools to study aging for many centuries. Of course, the need was mostly practical — to calculate mortality risks to estimate the cost of insurance. The pioneer in this field is Gompertz who in 1825 published a paper about his empirical observation that mortality risks increased exponentially with age, and were doubling roughly every 8 years. The so-called negligible senescent species are expected to defy this law, and a recent work by (Ruby et al. 2018) at Calico is a great example that the importance of actuarial research in aging is still high. They observed that the naked mole rat defied the law of mortality, and did not show any exponential growth of mortality during almost 40 years of constant observations.

After wrapping up my locomotor experiments and data analysis, I started looking into the statistics of mortality. Specifically, there was an interesting observation that mortality fitted to the Gompertz law would produce two parameters: the rate of aging and the initial mortality rate, which would be inversely correlated with each other. Hence, naively, an increased initial mortality causes a slower aging rate afterwards, and vice versa. This was and still is somewhat counterintuitive to me. In 1960, in the Science paper by Strehler and Mildvan, they put forward an interesting explanation of this relationship based on statistical physics and the Maxwell-Boltzmann distribution, which was subsequently called the Strehler-Mildvan correlation. They considered that relation fundamental, and expected any new quantitative theory of aging to be able to explain this.

Strehler’s logic was somewhat recursive and circular. They would introduce a variable, then rewrite an identity relation, and show that the parameters would correlate. Though, after a thorough disentanglement of their derivations, I’d end up with the original Gompertz law. There was no explanation to the law, it was simply postulated from their practical observations. Indeed, I could reproduce those observations and I had them at hand. The problem was that the correlation would appear even when I would simply resample the same mortality experiment a few times. The noise would produce the same perfect correlation between the aging rate and the initial mortality. No changes in biology, hence it must have been an artifact of the fit.

I analyzed some mortality data of nematodes, and for a long time it didn’t make much sense to me. Overall, the fit would be almost perfect (with the exception of a slight deviation in later ages, the so-called late-life mortality plateau). I would plot the rate of aging against the logarithm of the initial mortality, and indeed obtain a decent, suspiciously perfect correlation between the two parameters. It didn’t make much sense because there was nothing special about the data, in most cases I would stare at the experimental survival curves, and would not be able to tell any difference up to the noise level. At the same time, the Gompertz parameters would form a nice inversely correlated cloud, with correlation coefficients above 0.9. What I had learnt by then was that if you saw something like that in biology, most likely, you were doing something wrong or there was a technical artifact or another issue. After some reading and re-reading the paper by Strehler. Why would be a two-parametric curve effectively turn into a single-parametric one? Also, why would a population with a low or absent initial mortality age faster afterwards? Especially given that for humans the age gap between the two would be closer to 60 years. This counter-intuitiveness somehow resembled “a spooky action at a distance”. On top of that, for practical reasons of distinguishing the effects of potential anti-aging interventions it was important to be able to tell the difference between reducing the background/initial mortality and reducing the rate of aging.

For the Gompertz law, the two fitting parameters are nonlinearly related. Normally, the noise in parameter estimates would not correlate if they are independent. And, also, the aging rate would be exponentiated, whereas the initial mortality would not be. We spent some time with Peter and Leonid Ieronimovich trying to make sense of those observations. And, indeed, if one analytically writes down the objective function for a Gompertz fit, one would formally obtain a set of two equations. And the mathematical problem would be technically well-defined. The problem was that due to the extreme steepness of the survival curves, the system of equations would turn degenerate, and only one of them would stay independent. After a proper reparametrization, we figured out that the manifold formed by those degenerate solutions would coincide with the equation for a constant mean lifespan. Basically, almost any pair of parameters in the Gompertz fit producing a lifespan curve with the same mean lifespan would be a decent solution for the problem. Hence, when some experimental or sampling noise is present, one would, from time to time, converge to slighly different estimates of the parameters — and after sampling enough points, one would reproduce the Strehler-Mildvan correlation. We ended up publishing a math-heavy technical paper on this (Tarkhov et al. 2017).

Basically, we observed overfitting of a set of points to a two-parametric curve, whereas it is only well-defined for a single parameter — the mean lifespan. It was a somewhat sobering lesson for me — I knew from physics and basic ML that having too many parameters in the model is not great for its generalizability. Though, I didn’t think that having two parameters is already too many! The steeper the survival curves, the stronger that correlation is present. A characteristic parameter is the ratio of the mean lifespan to the mortality rate double time (MRDT). For humans, it’s 60-80 years over 8, i.e. it’s closer of order of 8-10. For worms, both the MRDT and mean lifespan are closer to 1 month, hence the ratio is closer to 1, and the overfitting is weaker.

Of course, there is more complexity in this story. Some people would try to parametrize survival curves with more parameters, they would introduce the baseline constant mortality rate, or add more stages to lifespan curves. But, overall, the logic would still hold — the survival curves are inherently very steep, and any additional parameters on top of the mean lifespan would be hard to resolve.

P.S. I’ll later get back to more topics related to actuarial works. One is the late-life mortality plateaus. Two is the temporal scaling of mortality curves in C.elegans nematodes. The third one is related to the historical trends of mortality parameters across decades and centuries — it would correspond to the rectangularization of mortality curves, and there is indeed some systematic shift present (even after taking into account all artifacts and degeneracies).

References

  1. A.E. Tarkhov, L.I. Menshikov, P.O. Fedichev. Strehler-Mildvan correlation is a degenerate manifold of Gompertz fit. J Theor Biol 416, 180–189 (2017). 10.1016/j.jtbi.2017.01.017
  2. B.L. Strehler, A.S. Mildvan. General Theory of Mortality and Aging: A stochastic model relates observations on aging, physiologic decline, mortality, and radiation. Science 132, 14-21 (1960). 10.1126/science.132.3418.14
  3. B. Gompertz. On the nature of the function expressive of the law of human mortality and on a new model of determining life contingencies. Phil. Trans. R. Soc. 115, 513–585. (1825). 10.1098/rstl.1825.0026
  4. J.G. Ruby, M. Smith, R. Buffenstein. eLife 7, e31157 (2018). 10.7554/eLife.31157
  5. A.M. Olovnikov. Printsip marginotomii v matrichnom sinteze polinukleotidov [Principle of marginotomy in template synthesis of polynucleotides]. Dokl Akad Nauk SSSR. 1971;201(6):1496-9. Russian. PMID: 5158754.
  6. N.N. Mushkambarov. Gerontologiya in polemico [Gerontology in Polemics]. Moscow: Meditsinskoe informatsionnoe agentstvo (2011). Russian.
updated_at 24-12-2024