Implied timescales

FU Berlin, Arnimallee 6, 14195 Berlin, Germany

Summary: Implied timescales refers to the relaxation timescales of a molecule implied by a Markov model transition matrix estimated at a lag time $\tau$. Since $\tau$ is a model parameter but the relaxation timescales is a physical property of the simulated system, it is expected that the implied timescales are independent of $\tau$. This is not the case for very small values of $\tau$, where the eigenvector approximation error and the spectral error of the Markov model are large; nor it is the case for very large values of $\tau$, where the estimation is dominated by numerical errors. For a good state space discretization, however, an intermediate range of $\tau$ values can be found in which the largest implied timescales should be approximately constant. Computing the implied timescales and plotting them is thus a useful tool to decide for a lag time $\tau$ at which the Markov model will be estimated. Testing whether the implied timescales are constant over a range of $\tau$ within statistical error is also a weak version of making a Chapman-Kolmogorow test.

One of the most interesting kinetic properties of molecular systems are the relaxation timescales of the system. They can be both experimentally accessed via relaxation or correlation functions that are measurable with various spectroscopic techniques [3][1][7][8], but can also be directly calculated from the Markov model eigenvalues as implied timescales: $$t_{i}(k\tau_{0})=-\frac{k\tau_{0}}{\ln\lambda_{i}(k\tau_{0})}. \:\:\:\:(1)$$ Where $\tau_{0}$ is some arbitrary reference lag time and $\tau=k\tau_{0}$ is the lagtime used. [14] has argued: if dynamics are exactly Markovian, then $\lambda_{i}(k\tau)=\lambda_{i}(\tau)^{k}$, and hence: $$t_{i}(k\tau_{0})=t_{i}=-\frac{\tau_{0}}{\ln\lambda_{i}(\tau_{0})}. \:\:\:\:(2)$$ must be independent of the lagtime chosen. Ref. [14] has therefore suggested to test if the slowest implied timescales, $t_{i}(k\tau)$, are approximately constant in the lagtime $k\tau$. Note that this is not a sufficient test of Markovianity, because it does not test whether the eigenvectors are also independent of $k\tau_{0}$. However, it has been empirically observed that constant timescales are a very strong indicator of having an MSM which approximates the underlying dynamics well. Hence, implied timescale tests are frequently used in order to select between different discretizations (clusterings) of state space.

If the discretized dynamics are not exactly Markovian, how do the implied timescales behave in $k\tau_{0}$? In [11], we have derived a tight bound for the slowest relaxation rate $\kappa_{2}=t_{2}^{-1}$. From this bound follows that the relative error of the largest implied timescale is bounded by: $$\begin{aligned} \frac{\hat{t}_{2}-t_{2}}{t_{2}} & \le & \frac{\ln\frac{1}{\alpha}}{\frac{\tau}{t_{2}}+\ln\frac{1}{\alpha}} \:\:\:\:(3)\end{aligned}$$ Where $\alpha=\langle\psi_{2},\hat{\psi}_{2}\rangle_{\mu}$ is the discretization quality with respect to the second propagator eigenfunction. In simple words, if $\alpha=1$, the state space discretization resolves the slowest process perfectly, while if $\alpha=0$, the slowest process is completely concealed by the discretization. For the limit $\frac{\tau}{t_{2}}\gg\ln\frac{1}{\alpha}$, i.e. when either the state space discretization is very good, or the lagtime is very large, the error becomes $$\begin{aligned} \frac{\hat{t}_{2}-t_{2}}{t_{2}} & \lessapprox & \frac{t_{2}}{\tau}\ln\frac{1}{\alpha}. \:\:\:\:(4)\end{aligned}$$ We observe that: (1) the implied timescales are well approximated if the state space discretization is very good ($\alpha=1\rightarrow\ln\alpha^{-1}=0$), (2) the implied timescale estimate converges towards the true implied timescale when the lagtime $\tau$ is increased. Unfortunately this convergence is very slow, with $\tau^{-1}$. Following [6], we can derive that also for all other relaxation processes, the corresponding implied timescale converges to its true value when the discretization quality increases or the lagtime $\tau$ increases:

$$\lim_{\tau\rightarrow\infty}\left|t_{j}(\tau)-\hat{t}_{j}(\tau)\right|=0, \:\:\:\:(5)$$

and also

$$\lim_{\delta_{j}\rightarrow0}\left|t_{j}(\tau)-\hat{t}_{j}(\tau)\right|=0. \:\:\:\:(6)$$

where $\delta_{j}=1-\alpha_{j}$ is the projection error of the state space discretization with respect to the $j$th dynamical process. This fact has been empirically observed in many previous studies [14][13][4][5][2][9][10].

From the mathematical facts above, the following rationale to assess the quality of the state space discretization follows:

  1. For a given state space discretization, estimate a series of transition matrices $\mathbf{T}(\tau_{k}=k\Delta t)$, where $\Delta t$ is the time step between saved trajectory frames and $k$ is a variable integer, using the methods described in lecture_ml_nonrev and lecture_ml_rev.

  2. Compute the $m$ largest eigenvalues of $\mathbf{T}(\tau_{k})$, and from these the $m$ slowest implied timescales $t_{i}(\tau_{k})$ depending on lag time $\tau_{k}$.

  3. When the implied timescales $t_{i}$ reach an approximately constant value for lagtimes $\tau_{k}$, the state space discretization is sufficiently good to resolve the dynamics in these slowest processes. Usually, it is also expected that the lagtimes for which this approximate constant value is reached are significantly smaller than the timescales $t_{i}$ of interest.

  4. Select the minimal lagtime $\tau$ at which $t_{i}$ are approximately constant, and use $\mathbf{T}(\tau)$ as Markov model.

Two notes of caution must be made at this point: (1) The argument above does not include the effect of statistics and is thus strictly only valid for the limit of good sampling. In many practical cases, statistics are insufficient and the implied timescales do not show the expected monotonous behavior that permits the quality of the discretization to be assessed. In this case, additional sampling is needed. (2) Observing convergence of the slowest implied timescales in $\tau$ is not a strict test of Markovianity. While Markovian dynamics implies constancy of implied timescales in $\tau$ [9][13], the reverse is not true and would require the eigenvectors to be constant as well. However, observing the lag time-dependence of the implied timescales is a useful approach to asses the quality of the discretization, and to choose a lag time $\tau$ at which $\mathbf{T}(\tau)$ shall be estimated, but this model needs to be validated subsequently (see lecture_chapman_kolmogorow).

Figure 1: Illustration of the eigenfunction approximation error $\delta_{2}$ on the slow transition in the diffusion in a double well (top, black line). The slowest eigenfunction is shown in the lower four panels (black), along with the step approximations (green) of the partitions (vertical black lines) at $x=50$; $x=40$; $x=10,20,...,80,90$; and $x=40,45,50,55,60$. The eigenfunction approximation error $\delta_{2}$ is shown as red area and its norm is printed.