Mathematical Modeling of the Aging Process

Mathematical Modeling of the Aging Process

Axel Kowald (Medizinisches Proteom Center (MPC), Ruhr-Universität Bochum, Germany)
Copyright: © 2009 |Pages: 19
DOI: 10.4018/978-1-60566-076-9.ch018
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Aging is a complex biological phenomenon that practically affects all multicellular eukaryotes. It is manifested by an ever increasing mortality risk, which finally leads to the death of the organism. Modern hygiene and medicine has led to an amazing increase in average life expectancy over the last 150 years, but the underlying biochemical mechanisms of the aging process are still poorly understood. However, a better understanding of these mechanisms is increasingly important since the growing fraction of elderly people in the human population confronts our society with completely new and challenging problems. The aim of this chapter is to provide an overview of the aging process, discuss how it relates to system biological concepts, and explain how mathematical modeling can improve our understanding of biochemical processes involved in the aging process. We concentrate on the modeling of stochastic effects that become important when the number of involved entities (i.e., molecules, organelles, cells) is very small and the reaction rates are low. This is the case for the accumulation of defective mitochondria, which we describe mathematically in detail. In recent years several tools became available for stochastic modeling and we also provide a brief description of the most important of those tools. Of course, mitochondria are not the only target of modeling efforts in aging research. Therefore, the chapter concludes with a brief survey of other interesting computational models in this field of research.
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What Is Aging?

Looking at the enormous rise of average human lifespan over the last 150 years, one could get the impression that modern research actually has identified the relevant biochemical pathways involved in aging and has successfully reduced the pace of aging. Oeppen and Vaupel (2002) collected data on world wide life expectancy from studies going back to 1840. Figure 1 shows the life expectancy for males (squares) and females (circles) for the countries that had the highest life expectancy for the given year. Two points are remarkable. Firstly, there is an amazingly linear trend in life expectancy that corresponds to an increase of 3 months per year (!) and secondly there is no leveling off observable.

Figure 1.

Male (blue squares) and female (red circles) life expectancy in the world record holding country between 1840 and 2000 based on the annual data of countries world wide (reproduced with permission from Oeppen & Vaupel, 2002).


These impressive data suggest strongly that lifespan will also continue to rise in the next years, but it does not show that the actual aging rate has fallen during the last century. Aging can best be described as a gradual functional decline, leading to a constantly increasing risk to die within the next time interval (mortality). The Gompertz-Makeham equation (Gompertz, 1825; Makeham, 1867),, describes how the exponential increase of mortality depends on intrinsic vulnerability (I), actuarial aging rate (G) and environmental risk (E). All living organisms have a base mortality caused by environmental risks, but it is the aging rate, G, which causes human mortality to double approx. every 8 years. From this equation we can derive the following expression for the survivorship function: 978-1-60566-076-9.ch018.m01 As expected we see that the number of remaining survivors depends on all three parameters and consequently a change of the average life expectancy (time until 50% of the population has died) can be caused by a modification of any of those parameters. This point is also discussed in more detail by (Kowald, 2002). And indeed, analyzing the survivorship data of the last 100 years more closely, it becomes clear that the aging rate, G, remained constant. The enormous increase in life expectancy was achieved exclusively by changes of intrinsic vulnerability and environmental risk!

Because of the drastic social, economical and political consequences that are brought about by the demographic changes of the age structure of the population, it is now more important than ever to understand what constitutes the biochemical basis for a non-zero aging rate, G. Systems biology might help to achieve this goal.


Why Is Aging A Prime Candidate For Systems Biology?

Evolutionary theories of the aging process explain why aging has evolved, but unfortunately they don’t predict specific mechanisms to be involved in aging. As a consequence more than 300 mechanistic ideas have been developed (Medvedev, 1990), each centered around different biochemical processes. This is probably due to the fact that even the simplest multicellular organisms are such complex systems that many components have the potential to cause deterioration of the whole system in case of a malfunction. Figure 2 shows a small sample of the most popular mechanistic theories. The spatial arrangement of the diagram intends to reflect the various connections between the different theories. And it is exactly the large number of interactions that makes it so difficult to investigate aging experimentally and renders it ideal for systems biology. To understand this we will look at a few examples.

Figure 2.

Graphical representation of some mechanistic theories of aging. The topology of the diagram reflects logical and mechanistic overlaps and points of interaction between different theories.


Key Terms in this Chapter

Systems Biology Workbench: The Systems Biology Workbench (SBW) is a software systems that enables different modeling programs to communicate with each other and provide or use specialized analysis services. In this way SBW acts as broker for services like deterministic and stochastic simulation engines, stability and bifurcation analysis, model optimization and graphical model building. Popular tools that are SBW aware are among others JDesigner, CellDesigner and Dizzy.

Mitochondria: Cellular organelles present in most eukaryotic cells that are important for calcium homeostasis, apoptosis and energy production. Mitochondria are endosymbionts and probably derived from purple bacteria. A remnant of this origin is the small circular mitochondrial DNA (mtDNA) that is at the center of the mitochondrial theory of aging.

Aging: A biological phenomenon observed in most animals leading to increasing functional impairment and constantly rising mortality rate. Age related changes can be observed at intracellular, tissue and organismic level. Many theories about the mechanism of the aging process exist, but the details are currently still unresolved.

Life Expectancy: Time until 50% of a cohort of newborn individual have died. Also known as average life-span, although technically it is the median life-span. The life expectancy for humans in industrialized countries is currently between 75 and 80 years, and for women 2-3 years higher than for man.

Dizzy: A stochastic simulation tool written in Java. Models can be defined in systems biology markup language (SBML) or a proprietary language and simulated using various stochastic and deterministic algorithms. The GUI and the core engine are separate modules so that Dizzy can also be used for batch calculations on a computer cluster.

Disposable Soma Theory: Popular theory about the evolution of the aging process. Aging is explained as the result of an optimal resource allocation between reproduction and self-maintenance. That means, aging itself has no selection advantage, but is a side product of another selected trait. Species specific life-spans are readily explained by different environmental mortalities.

Stochastic Modeling: A modeling framework that takes care of microscopic random fluctuations and the discreteness of molecules. Stochastic models explicitly calculate the change of the number of molecules of the participating species during the time course of a chemical reaction. The first exact stochastic simulation algorithms were developed by Gillespie (1977) and are now part of several modeling tools. Stochastic simulations are normally more time consuming than deterministic simulations via differential equations.

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