Aging is a fundamental aspect of living systems that undergo a progressive deterioration of physiological function with age and an increase of vulnerability to disease and death. Living systems, known as complex systems, require complexity in interactions among molecules, cells, organs, and individuals or regulatory mechanisms to perform a variety of activities for survival. On this basis, aging can be understood in terms of a progressive loss of complexity with age; this suggests that complexity in living systems would evolve with age. In general, aging dynamics is mathematically depicted by a survival function, which monotonically changes from 1 to 0 with age. It would be then useful to find an adequate survival function to link aging dynamics and complexity evolution. Here we describe a flexible survival function, which is derived from the stretched exponential function by adopting an age-dependent exponent. We note that the exponent is associated with evolving complexity, i.e., a fractal-like scaling in cumulative mortality. The survival function well depicts a general feature in survival curves; healthy populations show a tendency to evolve towards rectangular-like survival curves, as examples in humans or laboratory animals. This tendency suggests that both aging and complexity would evolve towards healthy survival in living systems. Our function to link aging with complexity may contribute to better understanding of biological aging in terms of complexity evolution.
Living systems are known as open self-organizing complex adaptive systems [1] and must often cope with hostile environmental conditions for survival [2]. To maintain their far-from-equilibrium (living) state, living things should exchange matter, energy, and information from their surroundings and further adapt themselves to genetic and environmental fluctuations [1,2]. To perform a variety of activities for survival, living systems require complexity in interactions among molecules, cells, organs, and individuals or regulatory mechanisms [3]. A hallmark of living systems is their extraordinary complexity in physiological function [3]. The concept of complexity, derived from the field of nonlinear dynamics in physics and mathematics, is useful to measure the output of complex physiologic processes in biology and medicine [3]. Healthy living systems with complexity are characterized by adaptive interactions of multiple control mechanisms that enable each individual to adapt to the exigencies and unpredictable changes of everyday life [3]. Generally, aging can be understood as a progressive loss of complexity with age in physiological systems [3]. The formulation of survival or mortality curves is essential for the quantification of aging dynamics to all scientists who study aging, such as demographers, biologists, and gerontologists. A single survival curve reflects a variance in survival probability (equally, survival function and survival rate) as a function of age. Despite recent physical approaches to population biology [4][5][6], a general picture of aging dynamics associated with complexity evolution of living systems remains elusive.
Many mathematical models for survival curves have been proposed (see recent reviews in refs. [7,8]). One of the fundamental mortality laws is the Gompertz law [9], in which the mortality rate increases roughly exponentially with increasing age at senescence. However, it seems to be obvious that the human mortality rate does not increase according to the Gompertz law at very old ages [8,10] and the deviation from the Gompertz law remains a great puzzle to demographers, biologists, and gerontologists. Many other mathematical models such as the Weibull, the Heligman-Pollard, the Kannisto, the quadratic, and the logistic models yet provide poor fit to the empirical mortality patterns at very old ages [8]. There are still needs for an appropriate mathematics for survival (or mortality) curves with simplicity, efficiency, and flexibility in complexity analysis [11][12][13][14]. In previous works [15][16][17], we put forward a universal survival function, which is derived from the stretched exponential function [18][19][20][21]. In this study, we address that this function enables us to connect aging dynamics to complexity evolution.
The stretched exponential function is widely used to describe complex dynamics in physics and biology [18][19][20][21]. In physics, relaxation is an aging process in which a system gradually changes from a far-from-equilibrium (living) to an equilibrium (dead) state. Structural relaxation of a glassy state towards a metastable equilibrium amorphous state is often referred as “physical aging”, which generally exhibits nonexponential relaxation [20,21]. The temporal behavior of the response function s(u) can be described by the stretched exponential or the Kohlrausch-Williams-Watts (KWW) function [22][23][24] (sometimes called the Weibull function [25]). This function has a general form of s(u) = exp(-u β ) (β > 0) where s(u) is the measurable quantity decreasing with age u (= x/α) where real age x can be rescaled with a characteristic life α taken at s(α) = exp(-1) ≈ 0.3679 [15]. The KWW function appears in many complex systems from soft matter systems, such as glass-forming liquids and amorphous solids [26,27], to astrophysical objects [28]. The KWW function is typically classified as the “stretched” exponential for 0 < β < 1, the “compressed” exponential for β > 1, and the “simple” exponential for β = 1. The nonexponential nature (β ≠ 1) is known to be related to the “dynamic heterogeneity” or the “fractal time” of the system [29]. In biology, the typical survival curves, s(u), for humans and animals fall into three main types, usually known as Type-I, -II, and -III curves [30]. Type-I survival curves slightly change at early and middle ages and then suddenly decline at late ages, as seen for long-lived humans. Type-II curves almost linearly decease with age, as seen for short-lived birds. Type-III curves quickly decrease at early ages, as seen for most plants. Interestingly, Type-I survival curves resemble the compressed exponential curves (β > 1). Type-II curves are similar to the standard exponential curves (β = 1). Type-III curves correspond to the stretched exponential curves (0 < β < 1). In this study, we describe a new survival function, which is well adaptable to biological survival curves by modifying the classic KWW function.
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