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Edited by: Mark A. LaBarge, Irell and Manella Graduate School of Biological Sciences, City of Hope, United States

Reviewed by: Gary An, University of Chicago, United States; David Robert Grimes, Queen's University Belfast, United Kingdom

This article was submitted to Molecular Medicine, a section of the journal Frontiers in Cell and Developmental Biology

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Here we present a theoretical and mathematical perspective on the process of aging. We extend the concepts of physical space and time to an abstract, mathematically-defined space, which we associate with a concept of “biological space-time” in which biological dynamics may be represented. We hypothesize that biological dynamics, represented as trajectories in biological space-time, may be used to model and study different rates of biological aging. As a consequence of this hypothesis, we show how dilation or contraction of time analogous to relativistic corrections of physical time resulting from accelerated or decelerated biological dynamics may be used to study precipitous or protracted aging. We show specific examples of how these principles may be used to model different rates of aging, with an emphasis on cancer in aging. We discuss how this theory may be tested or falsified, as well as novel concepts and implications of this theory that may improve our interpretation of biological aging.

The connection between one's chronological age and biological age is something that we all perceive. In a sense, it is the difference between the age you “feel” and the age you

From a mathematical perspective, biological dynamics as they relate to aging are frequently modeled as periodic or oscillating “clocks” describing, for example, circadian rhythms (Klerman and Hilaire,

Here we investigate a mathematical model of biological space-time which includes the effects of time dilation and contraction resulting from accelerated or decelerated biological clocks, which may provide a new theoretical foundation and perspective on rates of aging. The principle assumption of our model is that a dynamic biological process may be represented as the motion of a (massless) point along a trajectory on a manifold. We then investigate the consequence of time dilation and contraction in terms of accelerating or decelerating motion of the point along the trajectory, and relate these concepts to rates of biological aging, with a particular focus on aging in cancer. In particular, we put forth the hypothesis that a biological space-time may be used to model aging from an arbitrary biological viewpoint relative to a common frame of reference. A consequence of this hypothesis is that precipitous (faster than chronological time) or protracted (slower) aging may be modeled as relativistic corrections of dilation or contraction of time along the trajectory on the manifold in which the biological aging process occurs.

To the best of our knowledge, only a few groups have proposed similar concepts. Bailly and colleagues (Bailly et al.,

This manuscript is structured as follows: first we describe the mathematical objects which we use to define biological space-time. After defining the meaning of relativistic dynamics in this context, we show how relativistic corrections of dilation and contraction of time can be used to model precipitous or protracted aging. We then show examples of how these principles may be used to model aspects of the aging process with a particular emphasis on aging in cancer. We discuss the implications of this theory, including criteria that may be used to test or falsify the theory and suggest novel biological quantities that may improve our interpretation of biological aging.

Biology, and biological processes, are measured and observed in our conventional notion and understanding of physical space. Cells, tissues, and organisms move and change in a physical space that we can measure with length and time scales in conventional units. However, we may also consider the functional, or phenotype space in which biological processes can be represented as locations in the space. We refer to movement in a biological space as a sequence of locations in the space that form a trajectory. These general concepts have been used to characterize biological states such as hematopoietic differentiation, where 2- or 3-dimensional representations of biological space are constructed with dimension reduction techniques applied to high-dimensional single cell RNA-sequencing data (Mojtahedi et al.,

In order to provide a conceptual picture of our mathematical framework, we imagine the space related to a biological process identified by the index

Given a subset _{i} of a topological space (i.e., a set in which at each point it is possible to associate a neighborhood) _{i}-dimensional chart is an injective (one-to-one) function _{i} on _{i} spatial coordinates _{i, 1}∪_{i, 2}∪… is the whole space _{i}-dimensional differential manifold. For a more detailed discussion see (Tu,

A trajectory, or curve, γ_{i} on

Here _{i} represents the time evolution of the _{i}(_{i}(

The trajectory γ_{i} on the manifold _{i}(_{i}) and ends at _{i}(_{i}), where _{i}, ^{2} identified by the two unit vectors _{x} and _{y}.

Given the manifold ℝ^{+}, we now define the _{i} as the (_{i} + 1)-dimensional Lorentzian manifold given by the Cartesian product (the set of all ordered pairs) (see Tu,

A point _{i} is then identified by a set of _{i} + 1 coordinates

where _{i, μν} is the metric tensor of the

Since a large number of biological processes take place in the body of an individual, we assume the existence of several manifolds which may be indexed

A point _{1}, _{2}, …, _{N}) where each

where _{i} on

In analogy with Equation (2), the biological space-time 𝔐 for all biological processes occurring in a body of an individual is then defined by the

We consider the Cartesian product _{1} × 𝔐_{2} × ⋯ × 𝔐_{N} because in the latter case it is unclear how all time variables, one per each submanifold 𝔐_{i}, will combine together and define the time on the resulting manifold 𝔐. A point

The idea for which time flows at different rates for different biological processes is now mathematically modeled by introducing a set of projection maps Π_{i} such that:

where the spatial part _{i} on _{i} on the _{i} (Figure

A torus is shown as an example of a possible biological space-time manifold that may be decomposed into submanifolds. The time ℝ^{+} is represented by the vertical arrows. The torus is decomposed into the two circles (submanifolds) _{0} is mapped onto the points _{1} is mapped onto the points

We emphasize that the relationship between _{i} is not necessary linear. In fact, in the next section we will show that the time measured in a particular manifold depends upon the acceleration of the particle along a trajectory and it will produce a non-linear relation between _{i}.

We now provide a simple example which can be easily visualized in a three-dimensional space. We consider a manifold, the torus embedded in the three dimensional space ℝ^{3}, which can be decomposed into two circles, the two submanifolds. In this particular example, this is the only possible decomposition (Tu,

The torus, embedded in ℝ^{3}, is defined by the following set of equations:

and is decomposed into the two submanifolds

where, θ identifies the poloidal direction (along the orange vertical circle in Figure

where ^{+}, is then decomposed onto two trajectories:

where

which implies _{2} > _{1} for the same value of

This example shows that there can be motion on the torus and on one submanifold, but an absence of motion on the other submanifold.

The example of the torus illustrates how trajectories on submanifolds (e.g., representations of different biological dynamics) with different accelerations may be combined and interpreted as a single trajectory on a larger manifold, and vice versa. It must be noted that in this decomposition, the time

The fact that the Maxwell's equations are invariant under the Lorentz transformation implies that any inertial observer will measure light moving at the same constant speed

Although many investigators have posed the question of whether or not governing laws exist in biology (Ruse,

Although information in the context of biological processes can not be easily defined (Gatenby and Frieden,

As described in section 2.1, a point _{μν} is known. Moreover, the existence of such a parameter defines the non-relativistic and the relativistic limits in this context. The former occurs for velocities

Special relativity is universally recognized as a theory which describes properties of the ordinary space-time in which physical phenomena occur. In the case of a biological process, unlike the physical phenomenon, the biological space-time is not known a priori and it needs to be mathematically constructed. In this section we assume the existence of such a space and we consider the motion of two frames of reference whose coordinates are related by a set of coordinate transformations. In particular we investigate the dilation and contraction of biological time resulting from accelerated motions. This approach is used to model and to explore different rates of aging.

As shown by Levy-Leblond (

We consider two frames of reference ^{4} = ℝ × ℝ^{3}. In general, the motion of a particle is along a curvilinear trajectory, and the two frames of reference can be in a relative motion like the one represented in Figure

where γ > 1 is the Lorentz factor and

The time interval

This equation tells us how much time _{i}(_{i} introduced in Equation (8). In the case of an accelerating frame of reference, the phenomenon of time dilation is not reciprocal and the observer in the frame of reference

On the left) The particle (purple dot) is moving along a trajectory with velocity

Many markers have been used to define biological age. In our modeling framework, biological time is defined as the rate at which biological processes take place, as measured against chronological time. Therefore, our model characterizes one's biological age by the degree of dilation or contraction of time resulting from the acceleration or deceleration of biological processes that are associated with biological age.

One marker of biological age is the methlyation state of the genome, composed of varying degrees of hyper- or hypomethlyated states (Bocklandt et al.,

In order to explain the process of changing methylation states with age, and the use of methylation state as a surrogate marker of biological age with our model, we first consider the trajectory on a manifold which describes the methylation process of a living person and we assume that this trajectory is mapped onto a straight line in the Minkowski space. We now consider one frame of reference

If

The slowing down of the rate of demethylation is therefore interpreted as a decelerating frame of reference. We note that this model does not assume any specific functional form for the methylation trajectory. In fact, during the lifetime of a person, Equation (18) permits the acceleration or deceleration of age-related changes to the methylation state, which can be accentuated or modified in the context of cancer.

In this section we connect our model to different rates of aging between individuals, and discuss examples of precipitous or protracted aging within an individual.

We define aging as the functional and structural decline of an organism, resulting in an increasing risk of disease, impairment and mortality over time. At the molecular and cellular level several hallmarks of aging have been proposed to define common characteristics of aging in mammals (López-Otín et al.,

Ultimately, the rate of age-related decline varies depending on how genetic variation, environmental exposure and lifestyle factors impact these mechanisms. Consequently, age, when measured chronologically, is often not a reliable indicator of the body's rate of decline or physiological breakdown. Over the years, the idea of quantifying the “biological age" based on biomarkers for cellular and systemic changes that accompany the aging process have been explored (Levine,

As we derived in the previous section, an unambiguous time dilation effect requires an accelerating frame of reference. We now imagine a range of accelerations representing a distribution within a population between the values _{min} and _{max} (green region in Figure _{min} and _{max}, respectively. Any acceleration within this range will correspond to a curve in the green region. An interval of time Δ_{min} (pink region). Moreover, values of the acceleration much larger than _{max} can be used to define the arrest of biological time such as in a hibernating state or cryogenic freezing (blue region). The case of zero acceleration is given by Equation (16) and is not considered in this model context.

On the left) For an accelerating frame of reference a time interval Δ

We illustrate the contraction and dilation of time in biological space-time (i.e., on a submanifold) and how they can be related to the presence of a disease, we consider the following thought experiment involving Alice and Bob, two individuals with different rates of aging. For simplicity, we assume that their dynamics occur on a flat torus ^{2} plane under the identifications (^{2} with the identity map, and hence the dynamics on the manifold corresponds to the dynamics in the Minkowski space-time.

The flat torus

The point on the manifold corresponding to the birth of Alice and Bob is indicated in Figure _{x}, _{y}, _{z}). The dynamics of Alice is represented by the motion of a purple point in which we imagine to place a frame of reference

We now can apply the results obtained in the previous section to infer that the time in

Assuming the acceleration of Alice to lie within the range [_{min}, _{max}], her proper time

We now consider only Bob and we assume his frame of reference

where

Major pathologies, such as cancer, diabetes, cardiovascular disorders and neurodegenerative diseases have an impact on aging. Cancer and chemotherapy in particular are known to accelerate the aging process (Alfano et al.,

It is not surprising that the two cancer populations most affected by precipitous aging caused by cancer are survivors of childhood cancer who are typically exposed to intensive multi-agent therapy at a young age (Henderson et al.,

Frailty, characterized by a cluster of measurements of physical states, is the best described measure of aging in a population, and identifies individuals who are highly vulnerable to adverse health outcomes and premature mortality. Although frailty is not a perfect corollary with biological age, it is a measure of abnormal aging at a population level. In long-term survivors of childhood cancer [median age 33 years (range 18–50 years)], the prevalence of frailty (Rockwood and Mitnitski, _{c} < _{g}.

The normal process of aging is associated with chronic low grade inflammation and with cumulative oxidative stress, independently of disease. Inflammation and oxidative stress are critical responses in host defense and injury repair and are essential for normal body functions. However, with advanced age there is a loss of sensitivity in the injury-repair cycle leading to persistent chronic inflammation, and a natural decline in the endogenous anti-oxidant capacity leading to cumulative oxidative stress (Mittal et al.,

Chronic inflammation and oxidative stress are also common underlying factors of age-associated diseases. Inflammation, particularly chronic low-grade inflammation, has been found to contribute to the initiation and progression of multiple age-related pathologies such as type II Diabetes, Alzheimer's disease, cardiovascular disease and cancer (Mantovani et al.,

On the level of tissue homeostasis, the best characterized example is hematopoietic aging associated with chronic inflammatory signaling in the bone marrow microenvironment. In fact, the “age" of a young hematopoietic stem cell can be “reprogrammed" when transplanted into an aged or inflammatory environment (Kovtonyuk et al.,

In contrast to precipitous biological aging which corresponds to the contraction of biological time, protracted aging corresponds to the dilation of biological time. This can be illustrated by prolonged periods of near zero biological activity, for instance in freezing conditions or hibernation which is part of a continuum of biological and metabolic states (van Breukelen and Martin,

Here we have investigated a mathematical model of biological aging. We define a biological space-time by mathematically combining manifolds and submanifolds, and apply the principles of relativity to compare different rates of biological aging. We illustrate the concepts of precipitous and protracted aging as relativistic corrections of biological time with a mental illustration comparing the lifespans of two individuals, Bob and Alice. This analogy provides a framework to compare the rates of aging between individuals by determining their rate of acceleration as compared to a common frame of reference.

A critical component of our theory is the construction of “biological space-time,” hence a submanifold that represents a biological process. Following the work of (Levy-Leblond,

On the other hand, given our ability to determine the value of

Once the submanifolds are constructed, then it is always possible to combine them considering their Cartesian product defined by Equation (4). Given our ability to construct these submanifolds, the comparison between the dynamics along different trajectories can be done if and only if they do belong to the same submanifold. So for example, if we are interested in comparing different rates of aging of two individuals with respect to the methylation status, we will need to construct the methylation status manifold, evaluate the trajectories of their methylation status in time, and analyze the dynamics on each of them. In particular, the comparison between the accelerations along both trajectories may help in understanding the difference between the rate of aging of the two individuals. In this sense, the importance is the different dynamics along each trajectory and not the fact that there are two different trajectories.

Other groups have proposed a framework for biological time to explain biological rhythms and other oscillating or biological processes that repeat periodically (Bailly et al.,

A novel concept which naturally follows from our approach is the notion of

The shape of the manifolds which characterize biological space-time also affect our notion of distance, which requires the knowledge of the metric tensor of the manifold. Although a manifold is defined to be locally Euclidean, two biological processes or objects that are sufficiently different from each other (in space-time) may not be measured by using the conventional definition of Euclidean distance. The degree to which we must redefine our notion of a

In summary, we have proposed a mathematical framework and criteria that may be used to define and construct a biological space-time. We use this as a tool to model and study different rates of biological aging based on the concept of relativistic corrections of time due to the acceleration or deceleration of biological dynamics relative to a common frame of reference. We discuss some examples of biological processes that illustrate these concepts, provide criteria that may be used to test or falsify the theory, and discuss implications and novel hypotheses that are generated by this model.

RR and DM: Conception, design, and approval of final manuscript. All authors: Manuscript writing and editing.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We thank Leo D. Wang for a critical reading of the manuscript, and Jacob G. Scott for discussions that inspired and motivated this work. DM would like to thank G. Bocca for useful discussions.

By differentiating Equation (15) we obtain the transformations for the velocity

where _{x} = _{x} = _{x} =

where we have evaluated _{x} the

and the acceleration in

By integrating the above equation with respect to time and by imposing

By integrating once again with respect to time and by setting

The time