Quantitative characterization of biological age and frailty based on locomotor activity records

We performed a systematic evaluation of the relationships between locomotor activity and signatures of frailty, morbidity, and mortality risks using physical activity records from the 2003-2006 National Health and Nutrition Examination Survey (NHANES) and UK BioBank (UKB). We proposed a statistical description of the locomotor activity tracks and transformed the provided time series into vectors representing physiological states for each participant. The Principal Component Analysis of the transformed data revealed a winding trajectory with distinct segments corresponding to subsequent human development stages. The extended linear phase starts from 35−40 years old and is associated with the exponential increase of mortality risks according to the Gompertz mortality law. We characterized the distance traveled along the aging trajectory as a natural measure of biological age and demonstrated its significant association with frailty and hazardous lifestyles, along with the remaining lifespan and healthspan of an individual. The biological age explained most of the variance of the log-hazard ratio that was obtained by fitting directly to mortality and the incidence of chronic diseases. Our findings highlight the intimate relationship between the supervised and unsupervised signatures of the biological age and frailty, a consequence of the low intrinsic dimensionality of the aging dynamics.

In the following analysis we will assume that the TM W is irreducible and has distinct eigenvalues. The reasoning for such assumptions will be provided later. Under this assumptions W can be diagonalized: λ , 5 Where A k and B k are left (Σ i W ij A ki =λ k A kj ) and right (Σ j W ij B kj =λ k B ki ) eigenvectors corresponding to eigenvalue λ k . Note that the systems of left and right eigenvectors are the inverse for each other: δ δ . 6 To solve Eq. 3 we introduce , 7 and using Eqs. 5 and 6 rewrite Eq. 3 as λ , for which the solution is 0 λ , from which using Eqs. 6 and 7 we get where G ij (t) is the probability P(i,t|j,0) to find the system in state i at time t if the system originally was in state j at time 0 and P 0 is the initial distribution.
The assumption that W has distinct eigenvalues together with Eq. 4 imply that W has exactly one zero eigenvalue. Since the order of eigenvalues is arbitrary, we can state that λ 0, This equation holds for all i and j. For a given i let us choose a particular j such that ρ ik ≤ρ ij . Since all ρ ik and W kj for k≠j are non-negative by definition, the Eq. 11 becomes Reλ i ≤0.
According to Eq. 3, any equilibrium state is the right eigenvector corresponding to the zero eigenvalue. Since W has only one such eigenvector (up to scaling), we have a unique equilibrium distribution given by www.aging-us.com

12
The eigensystem has several interesting properties. From Eq. 9 and 10 we get G ij (+∞)=A 1j B 1i and the distribution at t=+∞ is ∞ . 13 For any initial distribution P 0 the corresponding P ∞ is an equilibrium state: and since equilibrium is unique P i ∞ = P i eq for any P 0 . From this and Eq. 13 we have

1
. 14 Using Eq. 4, for the right eigenvectors B k we get λ 0, and therefore Let as consider a discrete real-valued stochastic process x(t) having value x i when the system happens to be in state i. According to the Wiener-Khinchin theorem, the power spectral density S x (ω) for the x(t) is the Fourier transform of the autocorrelator Using the fact that R xx (τ) is an even real-valued function we obtain ω 2 τ cos ωτ τ ∞ . 17 Here we follow the common physical convention that the total power of the signal is given by Expanding the Eq. 16 we get Where P(i,t+τ|j,t) is the probability to find the system in state j at time t+τ if the system was in state j at time t and P(i,t) is the probability to find the system in state j at time t, with the evolution of the system starting from some state P 0 . From the definitions we have P(i,t+τ|j,t)=G ij (τ) for τ≥0 and P(j,t)=P j (t). Using this and Eq. 8 and 9 rewrite Eq. 18 as where τ>0 and the summation is done for each index from 1 to N. By rearranging and using Eq. 10 we get τ λ τ , , , , from which using Eq. 12 we finally obtain τ λ τ , τ 0.

19
Note that R xx (τ) is not dependent on the initial distribution P 0 , as it is expected for the system with equilibrium state. The integration of Eq. 19 using Eq. 17 is straightforward, and we get The Eq. 20 is valid for any irreducible diagonalizable TM W. In particular, some λ k may be complex. However, for the real-valued matrix W complex eigenvalues and corresponding eigenvectors always comes in complex conjugate pairs, which, together with Eq. 10, imply that S x (ω) is always real positive, as any PSD should be.
Due to time symmetry of the fundamental physical laws, for the systems in thermodynamic equilibrium the detailed balance assumption is hold: . 21 Biological organisms as a whole are not systems in thermodynamic equilibrium and the description of the motion using Markov chain model is only a rough www.aging-us.com approximation, so there are no a priori reasons to assume the detailed balance. However, experimentally the correlation between W ij P j eq and W ji P i eq is good, so it is interesting to see how S x (ω) looks under detailed balance assumption. . 29 The S x (ω) can be calculated under detailed balance assumption as follows: calculate the right eigensystem for W, scale the found eigenvectors B k using Eq. 29 and finally calculate S x (ω) using Eq. 27. The same procedure can be applied when the detailed balance assumption holds only approximately, as long as we drop the imaginary part of the found eigenvalues and right eigenvectors. Note that even when all eigenvalues are real, the Eq. 27 is not equivalent to Eq. 20 without the detailed balance assumption. In particular, scaling according to Eq. 29 is not enough for Eq. 28 to hold, which is required for Eq. 27 to be precise. www.aging-us.com