Data-driven approach to decomposing complex enzyme kinetics with surrogate models

The temporal autocorrelation (AC) function associated with monitoring order parameters characterizing conformational fluctuations of an enzyme is analyzed using a collection of surrogate models. The surrogates considered are phenomenological stochastic differential equation (SDE) models. It is demonstrated how an ensemble of such surrogate models, each surrogate being calibrated from a single trajectory, indirectly contains information about unresolved conformational degrees of freedom. This ensemble can be used to construct complex temporal ACs associated with a “non-Markovian” process. The ensemble of surrogates approach allows researchers to consider models more flexible than a mixture of exponentials to describe relaxation times and at the same time gain physical information about the system. The relevance of this type of analysis to matching single-molecule experiments to computer simulations and how more complex stochastic processes can emerge from a mixture of simpler processes is also discussed. The ideas are illustrated on a toy SDE model and on molecular-dynamics simulations of the enzyme dihydrofolate reductase.

Figure 1

(Color online) The (un-normalized) AC of four realizations, distinguished by different color shades in the plot, from four toy processes discussed in the text (left). The time lag $\delta t$ reported is in ps. The standard deviation of the AC curves measured in a trajectorywise fashion from a population of 100 trajectories. The different line shades denote different temporal samples sizes (see text). The computed standard deviation reflects the variance in a population of 100 empirically determined ACs.

Figure 2

(Color online) Average local diffusion function estimated (left axis) and free energy (right axis) as a function of order parameter. The population average of the estimated $C^2$ $\left({\text{Å}}^2/\text{ps}\right)$ is plotted to give a feel for the position dependence of this quantity; it is stressed that the average alone is not adequate to describe the dynamics here due to a type of dynamic disorder phenomena [1]. The free energy (kcal/mol) was computed by Arora and Brooks III using methods described in Ref. [7].

Figure 3

(Color online) Normalized local AC function of MD data (more jagged curves) and that predicted by surrogates. The total time series of length 1.2 ns was divided into separate blocks, each containing 400 observations spaced by 0.15 ps (the time lag $\delta t$ is reported in ps). These data were used to estimate a collection of surrogate models and a collection of MD ACs corresponding to US target points of $\Delta D_{\text{rmsd}}\approx -1.8$, 0.9, 1.9, and $3.1\text{Å}$. The AC corresponding to the estimate surrogate models is also plotted where the initial lag is normalized to unity to facilitate comparison (the raw units are shown in Fig. 4).

Figure 4

(Color online) The global (un-normalized) AC prediction. The procedure introduced here is used along with PDOD data to predict the long-time AC (the time lag $\delta t$ is reported in ps). The relaxation predicted by the individual surrogate (data shown in Fig. 3) are also shown to stress that the SDE parameters are not fixed, but evolving and any single surrogate cannot capture the richer long-time dynamics.

Figure 5

(Color online) Goodness-of-fit tests. The test of Ref. [24] was computed given the parameter estimate and the observed data. The median of each umbrella sampling window is reported.

Figure 6

(Color online) Stationary density/histogram prediction. The bars denote 1.2 ns MD data, the solid thick line denotes the mixture PDOD method (see text), the solid dotted line denotes the average PDOD model, and the thin lines denote the four local equilibrium densities used in constructing the mixture density.