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Non-Markovian properties and multiscale hidden Markovian network buried in single molecule time series.

We present a novel scheme to extract a multiscale state space network (SSN) from single-molecule time series. The multiscale SSN is a type of hidden Markov model that takes into account both multiple states buried in the measurement and memory effects in the process of the observable whenever they exist. Most biological systems function in a nonstationary manner across multiple timescales. Combined with a recently established nonlinear time series analysis based on information theory, a simple scheme is proposed to deal with the properties of multiscale and nonstationarity for a discrete time series. We derived an explicit analytical expression of the autocorrelation function in terms of the SSN. To demonstrate the potential of our scheme, we investigated single-molecule time series of dissociation and association kinetics between epidermal growth factor receptor (EGFR) on the plasma membrane and its adaptor protein Ash/Grb2 (Grb2) in an in vitro reconstituted system. We found that our formula successfully reproduces their autocorrelation function for a wide range of timescales (up to 3 s), and the underlying SSNs change their topographical structure as a function of the timescale; while the corresponding SSN is simple at the short timescale (0.033-0.1 s), the SSN at the longer timescales (0.1 s to ~3 s) becomes rather complex in order to capture multiscale nonstationary kinetics emerging at longer timescales. It is also found that visiting the unbound form of the EGFR-Grb2 system approximately resets all information of history or memory of the process.