Bayesian

The study introduces a mathematical framework to model the concentration of molecular components over time, accounting for degradation processes that can occur in distinct phases. The core of the model is represented by the following equation:

$$
begin{array}{ll}
text{C}_{text{it}} = & left{ begin{array}{ll}
text{C}_{text{i,t}=0} times e^{-lambda_{1i} times t} & text{if } t le t_{ix} \
text{C}_{text{i,t}=t_{ix}} times e^{-lambda_{2i} times (t – t_{ix})} & text{if } t > t_{ix}
end{array} right.
end{array}
$$

(4)

where (:{text{C}}_{it}) represents the concentration of each component (c) and marker (m) at time (t). (:{text{C}}_{text{i},text{t}=0}) denotes the initial concentration at time zero. (:{{uplambda:}1}_{text{i}}) and (:{{uplambda:}2}_{text{i}}) are the decay rate constants for the first and second phases, respectively. (:{text{t}}_{text{i}text{x}}) signifies the time at which the decay rate changes, and (:{text{C}}_{text{i},text{t}={text{t}}_{ix}}) is the concentration at this transition point, calculated as (:{text{C}}_{cm,t={t}_{ix}}={text{C}}_{text{i},text{t}=0}times:{e}^{-{{uplambda:}1}_{text{i}}times:{text{t}}_{ix}}).

The model integrates concentrations from two technical replicates (r) for each component. The decay rates ((:{{uplambda:}1}_{text{i}}) and (:{{uplambda:}2}_{text{i}})) and the transition time ((:{text{t}}_{text{x}})) are estimated for each component and marker combination across three biological replicates. The implementation utilized the Stan language via the R package Rstan⁶⁹, employing four autonomous Markov chain Monte Carlo (MCMC) chains. Each chain underwent 5000 warmup iterations followed by 10,000 sampling iterations. The model achieved effective sample sizes exceeding 500 and convergence parameters ((:widehat{R})) below 1.005 for all estimated parameters, indicating robust model performance.

To serve as potential time-sensitive indicators, termed “molecular clocks,” two specific molecular